Interior Estimates for Some Nonconforming and Mixed ... - math.umn.edu

The Pennsylvania State University The Graduate School Department of Mathematics

INTERIOR ESTIMATES FOR SOME NONCONFORMING AND MIXED FINITE ELEMENT METHODS A Thesis in Mathematics by Xiaobo Liu c 1993 Xiaobo Liu Submitted in Partial Fulllment of the Requirements for the Degree of Doctor of Philosophy October 1993

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CHAPTER 1 INTRODUCTION Finite element methods are widely used for approximating elliptic boundary value problems. Usually the accuracy of such numerical methods depend on both the smoothness of the exact solution and on the order of complete polynomials in the nite element space. To be specic, consider the Dirichlet problem for the Poisson equation  u = f in  (1.1) u = 0 on @  where  is a bounded polygonal domain in R2 (so that the nite element space can be constructed without error in approximating the boundary) and f is some given function. The standard nite element method for (1.1) consists of constructing a oneparameter family of continuous piecewise polynomials subspaces V h of the Hilbert space H 1 () and using the Ritz-Galerkin method to compute an approximation uh 2 V h . The standard error estimate gives

ku ; uhk1   C vinf k u ; vk1   Chmin(k r;1)kukr   2V h

(1.2)

where k  kr  is the norm on Hilbert space H r () and k is the order of complete polynomials in the nite element space V h . In order for the nite element solution to achieve the optimal convergence rate, the exact solution u must be suciently regular. Namely, if r  k + 1, then (1.2) will result in an O(hk ) order convergence rate, which is best possible for the degree of polynomials used. Otherwise a loss of accuracy will occur. In practice, it often happens that r < k + 1. For example, when  is a nonconvex polygon, the exact solution will generally have corner singularities, and one cannot

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expect u to be in H 2 (). So no matter how high the order of the nite element space V h is, the nite element approximation does not even achieve rst order convergence. The situation is even worse for the plane elasticity problem, described by a second order vector elliptic equation. In this case, the solution may not be in H 2 () even if  is a convex polygon 29] (under some boundary conditions). We also note that there are other important situations when the exact solution is singular or nearly so, even when the boundary is smooth, for example, in singular perturbation problems or problems with concentrated loads. In the examples mentioned above, the exact solution is smooth in a large part of the domain and the singularity is a local phenomenon. Therefore, it is natural to ask whether uh approximates u better where u is smoother. Interior error estimates address this question. Interior error estimates for nite element discretizations were rst introduced by Nitsche and Schatz 33] for second order scalar elliptic equations in 1974. They proved that for h suciently small
ku ; uhk1   C ; vinf k u ; v k + k u ; u k 1  h ; p  2V 0

h

1

1

(1.3)

for 0 b 1 b  (here A b B means that A  B) and any nonnegative integer p. Here C is a constant that is independent of u, uh, and h. This estimate says that the local accuracy of the nite element approximation is bounded in terms of two factors: the local approximability of the exact solution by the nite element space and the global approximability measured in an arbitrarily weak Sobolev norm on a slightly larger domain. The usual way to estimate ku ; uhk;p 1 is to use the fact that ku ; uhk;p 1  ku ; uhk;p , for which the estimate is available by using Nitsche's duality technique. The signicance of the negative norm is that, under some very important circumstances, one can prove higher rates of convergence in

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the negative norm than that in the energy norm. Therefore, better convergence rates may be obtained in the interior domain. But it does not imply that one can always recover the optimal convergence rate. For example, as a direct application of (1.3) and the standard convergence theory of the nite element method, it is easy to see that if linear Lagrange elements are used for the Poisson equation on an L-shaped domain with a smooth forcing function, then ku ; uhk1 0 is of O(h) for any interior region 0 . However, if quadratic Lagrange elements are used for the same problem, ku ; uhk1 0 is only of order O(h4=3), which is less than the optimal O(h2) rate (but better than the O(h2=3 ) global rate). This phenomenon is called the pollution e ect of the boundary singularity. In 1977, Schatz and Wahlbin extended the idea of 33] and established interior estimates in the maximum norm 35] for second order elliptic equations. They proved that r
ku ; uhk1 0  C ; (ln h1 ) vinf k u ; v k + k u ; u k (1.4) 1  h ; p  1 1 2Vh where k  k1 0 represents the usual maximum norm and r = 1 for linear elements in R2, r = 0, otherwise. This was later generalized to allow 0 to intersect the boundary of . Interior error estimates are important for other reasons as well. In some cases, mesh renement and post-processing schemes to improve the initial approximation can be designed by using the information obtained from a local analysis. In 1979 Schatz and Wahlbin 37], based on (1.4), gave a systematic mesh renement procedure for the nite element method for second order elliptic equations on polygonal domains and showed that optimal global convergence rates could be obtained. In 1983, they studied in detail the approximation of the standard nite element method for the singular perturbed second order elliptic equation, where a strong boundary

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layer eect exists 38], again utilizing the interior convergence theory. In 1985 Eriksson 24], 25] applied the local analysis method to the second order elliptic equations with singular forcing functions and designed an adaptive mesh renement scheme to obtain optimal convergence rates. They also generalized such methods to some time dependent problems 27]. Interior error estimates have also been used successfully to study a posteriori estimators. In 1988 Eriksson and Johnson 26] introduced two a posteriori error estimators based on local dierence quotients of the numerical solution. Their analysis was based on the interior convergence theory in 33]. Zhu and Zienkiewicz 45], 46] proposed several adaptive procedures for nite element methods based on smoothing techniques. In 1991, Babuska and Rodrguez 9] gave a complete study of these estimators by using the interior estimate results of Bramble and Schatz 12]. In 1992, Duran 22], 23] proved the asymptotic exactness of several a posteriori error estimators by Bank and Weisser 10] by applying the interior superconvergence results of Whiteman and Wheeler 42]. The interior convergence theory is reasonably well understood for standard nite element methods. For a comprehensive review, see 41]. But there are only few results in this area for mixed nite element methods. The diculty in obtaining interior estimates for mixed methods can be understood by considering how an interior estimate is usually obtained: rst the exact solution is restricted to a local domain and its projection is constructed then the dierence between the global nite element solution and the local projection of the exact solution is estimated via duality and energy arguments. For the interior analysis of a mixed method, there are two new aspects compared to that for a standard one: the coupling of local projections and the balancing of two dierent norms. The resolution of these problems depends on the specic mixed formulation. In 1985 Douglas and Milner 20] adapted the

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Nitsche-Schatz approach to the Raviart-Thomas mixed method for scalar second order elliptic problems. Their work took advantage of the so{called \commuting diagram property" 21] between the two discrete spaces. Recently, Gastaldi 28] obtained interior error estimates for some nite element methods for the Reissner{ Mindlin plate model. Her work is similar in spirit to that of Chapter 4. However it is for the Brezzi-Bathe-Fortin family of elements for the Reissner-Mindlin plate 14], for which the variational formulation is dierent. The \commuting diagram property" plays an important role in Gastaldi's work, but does not enter here. In this thesis we establish interior estimates for some nonconforming and mixed nite element methods. Our primary goal is the interior error analysis for the the Arnold-Falk element for the Reissner-Mindlin plate model 3]. Via the Helmholtz decomposition, the Reissner-Mindlin system can be transformed into an uncoupled system of two Poisson equations and a singularly perturbed variant of the Stokes system. Using a discrete Helmholtz decomposition theorem, the Arnold-Falk element can be viewed as combination of nonconforming linear elements for the Poisson equations and the MINI element 2] for the Stokes-like system. Therefore the interior analysis of the Arnold-Falk element requires analysis of the nonconforming piecewise linear nite element for the Poisson equation and of the MINI element for the Stokeslike system, and so we consider those problems, which are also of interest in their own right, rst. The thesis is organized as follows. Chapter 2 denes with some notation and derives interior estimates for the linear nonconforming nite element method for the Poisson equation. This result will be used later in Chapter 4 in the interior estimate of the Arnold-Falk element for the Reissner-Mindlin plate model. Because of the relative simplicity of this chapter, it also serves to review the standard procedure for obtaining interior error estimates. Chapter 3 gives interior error estimates in the

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energy norm for a wide class of nite element methods for the Stokes equations. In Chapter 4 we study the interior error estimate of the Arnold-Falk element for the Reissner-Mindlin plate model. First by adapting the theory of Chapter 3, we obtain the interior estimate for the Stokes-like system. This is later used to prove that the Arnold-Falk element achieves (almost) rst order convergence rate uniformly in the plate thickness t in any interior region. Note that rst order convergence cannot be achieved globally (for the soft simply supported plate), due to the existence of a boundary layer in the exact solution. This problem does not arise for the hard clamped boundary conditions considered in 3], since in that case the boundary layer is weaker, and global rst order convergence is achieved. Numerical results are given, which conrm the theoretical prediction. Finally, in the Appendices, we prove two technical results, one about approximation property of linear nite elements and the other about the regularity of the exact solution of the Stokes-like system. They are required in Chapter 4.

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CHAPTER 2 INTERIOR ESTIMATES FOR A NONCONFORMING METHOD 2.1 Introduction In this chapter, we rst introduce some standard notations and then take the Poisson equation as an example to study the interior error estimate for the nonconforming nite element method. The linear element will be the focus of the study and the result obtained here will be used in Chapter 4 in the interior estimate of the Arnold-Falk element for the Reissner-Mindlin plate model. We note that more general results can be obtained similarly. The technique used is a combination of those in 33] and 34]. Even though the method of getting interior estimates for nite element methods is well known (for a comprehensive review, see 41]), the result proven here, to the author's knowledge, is new. We mention that in 1990 Zhan and Wang 44] obtained interior estimates for a class of (compensated) nonconforming elements for second order elliptic equations. However, the method they considered there excludes most standard nonconforming methods, including the linear element we will study here. Overall, the structure of this chapter is quite similar to that in 33] for the continuous element. So this chapter can serve to review the standard procedure of obtaining interior estimates. Some dierences still exist: (1) in section 2.4, additional terms have to be taken into account due to the discontinuity of nite element functions across element edges (2) in section 2.5, the integration by parts technique, which is essential in Nitsche and Schatz's treatment for the continuous element 33, section 5],

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is not used. Instead, we use a method by Schatz in 34]. The remainder of this chapter is organized as follows. Section 2.2 presents notations and the model equation. Section 2.3 gives a brief introduction to the nonconforming nite element space and proves some of its properties. Section 2.4 derives an interior duality estimate and section 2.5 shows the nal result.

2.2 Notations and Preliminaries The notations used in this chapter (as well as the whole thesis) are quite standard. For those for Sobolev spaces, cf. Adams 1]. Let   R2 be a bounded domain with Lipschitz boundary @ . Lp() is the usual space consisting of p-th power integrable functions. W m p() will be the standard Sobolev space of index (m p) with norm denoted by k  km p  , for m 2 N. The fractional spaces can be dened by interpolation 32]. We shall use the usual L2based Sobolev spaces H s () and H s (@ ), s 2 R, with norms denoted by k  ks  and kks @ , respectively. Notation jjs  denotes the semi-norm of H s (). We will drop  and use H s to denote H s(), with norm k  km , whenever no confusion can

s is the completion of C01() in H s . arise. The space H For s  0, H ;s denotes the closure of C01() under the norm

kuk;s  = sup k(vuksv) : v2H s v6=0

The notation ( ) stands for both the L2 inner product and its extension to a pairing of H s and H ;s . The notation h i denotes the pairing of H s(@ ) and H ;s(@ ). We use boldface type to denote 2-vector-valued functions, operators whose values are vector-valued functions, and spaces of vector-valued functions. This is illustrated

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in the denitions of the following standard dierential operators: 1 + @2 div = @ @x @y





grad p = @p=@x @p=@y :

The letter C denotes a generic constant, not necessarily the same in each occurrence. Consider the boundary value problem

;  u = K ; div F u=0

in 

(2.2.1)

on @ :

(2.2.2)

In the above, we include div F on the right hand side since it appears in a reformulation of the Reissner-Mindlin plate equations for which we will study in Chapter 4. This plate model was the original motivation for the current investigation. The weak variational form is: Find u 2 H 1 such that (grad u grad v) = (K v) + (grad v F ) for all v 2 H 1 :

(2.2.3)

From the standard theory on elliptic boundary value problems (cf. 32]), we have:

Lemma 2.2.1. For a smooth , a given K 2 H k , and an F 2 H k+1, there is a unique solution u satisfying (2.2.1) and (2.2.2). Moreover,

kukk+2  C ; kK kk + kF kk+1


(2.2.4)

where C is independent of K , F , and u.

2.3 The Nonconforming P 1 Element The notations and denitions for nite element spaces used here follow closely those by Ciarlet 16]. For simplicity, we will assume that  is a polygonal domain.

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This is just to avoid explaining the construction of curved elements near the boundary @ . The theory of the interior estimate to be developed in this chapter, however, is independent of this assumption. By a triangulation of  we mean a set Th of closed triangles such that the intersection of any two triangles is either a common edge, a common vertex, or empty, S and such that  = K2Th K . For any K 2 Th , let hK be its diameter and K the radius of the largest inscribed disk inside K . Dene h = maxK2Th hK . We will assume that triangulation Th is quasi-uniform (cf. 31, page 141]), i.e., there are positive constants 1 and 2 independent of h such that

hK  1 h for all K stated. Dene

2 Th .

K hK

 2

This restriction carries over to the whole thesis unless otherwise

Wh = fw 2 L2 : wjT 2 P1(T ) for all T 2 Th w is continuous at midpoints of element edgesg

W h = fw 2 L2 : wjT 2 P1(T ) for all T 2 Th w is continuous at midpoints of element edges and vanishes at midpoints of boundary edgesg

Vh = fv 2 H 1 : vjT 2 P1 (T ) for all T 2 Th v is continuous at element verticesg:

Here P1(T ) is the set of linear functions on T . The sets Wh and Vh are the standard nonconforming linear nite element space and the conforming linear nite element space, respectively. For 0  , let

h (0 ) = fp 2 Wh j supp p   0g W

V h (0 ) = fv 2 Vh j supp v   0 g:

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If Gh b  is a union of triangles, let Wh (Gh ), W h (Gh ), and Vh (Gh ) be dened the same way as Wh , W h , and Vh , respectively. Let G0 and G be two concentric open disks with G0 b G b , i.e., G 0  G and G  . Then there is a positive number h0, such that for h  h0, the following properties hold. Superapproximation property. Let ! 2 C01(G0 ) and u 2 Wh . There exists a

h (G), such that function v 2 W

k gradh(!2 u ; v)k0 G  Ch; k! gradh uk0 G + kuk0 G


(2.3.1)

for C = C (G0 G !). Here for 2 Wh , gradh denotes the function with values in the space of piecewise constants that coincides element-wise with grad . Inverse inequality property. Let t be a nonnegative integer. Then there exists a set Gh , which is a union of triangles and satises G0 b Gh b G, such that

kukh1 G  Ch;t;1kuk;t G h

h

for all u 2 Wh

(2.3.2)

where the constant C is independent of h and u. Here kukh1 Gh = (k gradh uk20 Gh + kuk20 Gh )1=2 for u 2 Wh. The above superapproximation property is somewhat dierent from the one in 33] (cf. section 3.3). This is because a dierent approach will be used in the step of \interior error estimates" 33, section 5] (from that for the conforming elements). We mention that (2.3.1) was rst proved by Schatz 34] for the continuous linear element and the same proof can be carried over to the nonconforming element. For the sake of completeness, we include the proof. Proof of the superapproximation property. As ! 2 C01(G0), for h small enough, we can nd a set Gh , a union of triangles, such that G0 b Gh b G. By the standard

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approximation theory on nite element spaces (cf. 16, page 121]), the linear function vT which interpolates !2u at the midpoints of the edges of T satises

k grad(!2u ; vT )k0 T  CT hT k grad(!2u)k1 T  CT hT (k! grad uk0 T + kuk0 T ) for each T 2 Gh . Dene v in Wh (Gh ) by vjT = vT and extend it outside Gh by zero

h (G). Summing up inequalities of above type for each T and using so that v 2 W the fact that Th is quasi-uniform, we obtain (2.3.1). An inverse inequality like (2.3.2) was used in 33] for continuous elements, where it was stated that it could be obtained by using the inverse inequality kukt Gh  Chs;tkuks Gh , for 0  s  t, and Green's formula. It is unlikely that this approach can be easily adapted for discontinuous elements. In the following we give a proof that is independent of the specic nite element space. Proof of the inverse inequality. The proof uses a result by Schatz and Wahlbin 35, Lemma 1.1]. Let t  0 be an integer. Furthermore, let j , j = 1 : : : J , be disjoint open sets S with  = Jj=1 j . Then J X j =1

kuk2;t   kuk2;t  j

for all u 2 H ;t :

Based on the above inequality and the standard inverse inequality 16, page 112]

kukh1  Ch;1kuk0

for all u 2 Wh

it is easily seen that one need only prove

kuk0 K  Ch;tkuk;t K

for all u 2 P1(K ) and K 2 Th :

(2.3.3)

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To do so, we apply the scaling argument 16]. Let K^ be the standard reference triangle and FK an ane mapping from K^ into K . For any function v 2 L2 (K ), let v^(^x) = v(x), where x = FK (^x). Under FK , the set Wh (K ) will be mapped onto P1(K^ ), the space of linear polynomials on K^ . Using the equivalence of norms on a nite dimensional linear space, we obtain

ku^k0 K^  C ku^k;t K^

for all u^ 2 P1 (K^ )

(2.3.4)

with C independent of u^. By denition

ku^k;t K^ =

(^u v^)K^ : k v ^ k ^ t t K  ^ v^2H (K ) sup

(2.3.5)

v^6=0

We have (cf. 16, page 140]) (^u v^)K^  Ch;K2(u v)K and

t

kv^kt K^  Ch;K1; X h2Ki jvj2i K
 ChtK;1kvkt K 1 2

i=0

(2.3.6) (2.3.7)

with constant C depending only on the minimum angle of K . Substituting (2.3.6) and (2.3.7) into (2.3.5) yields

ku^k;t K^  Ch;Kt;1

(u v) = Ch;t;1kuk : ;t K K v2H t (K ) kv kt K sup v6=0

Since

kuk0 K  ChK ku^k0 K^

and the mesh is quasi-uniform, inequality (2.3.3) follows. The nite element approximation for (2.2.1) and (2.2.2) is:

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Find a uh 2 W h such that (gradh uh gradh v) = (K v) + (gradh v F ) for all v 2 W h : The following convergence theorem is well known. 3, Lemma 5.4].

(2.3.8)

See, for example

Lemma 2.3.1. Let K 2 L2 and F 2 H 1 . Assume that  is a convex polygon and u and uh are the solutions of (2.2.1) and (2.2.2), and (2.3.8), respectively. Then,

ku ; uhk0 + hk gradh (u ; uh)k0  Ch2; kK k0 + kF k1
:

(2.3.9)

The following estimate, which can be found in 18], will play an important role in our analysis.

Lemma 2.3.2. There is a constant C independent of h such that  Z  X  

T 2Th @T

uw 

  nT 

 Chkwk1 vinf k gradh (u ; v)k0  2H 1

for all w 2 H 1 u 2 W h + H 1

(2.3.10)

where nT is the outer normal of each triangle T .

Before we turn to the next section, we dene a semi-norm for linear functional L

h (G). on W kLkG = sup k gradL(v)vk : h (G) v2W

gradh v6=0

h

0G

We also want to point out that the results of this chapter require that the mesh size h to be suciently small (which is self-evident from the analysis involved). However, for the sake of simplicity, we may not mention it explicitly.

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2.4 An Interior Duality Estimate In this section we will derive an interior duality estimate. The method used here is parallel to that for the conforming method, but there are some additional terms, which measure the jumps of the discontinuous nite elements, to be taken care of. Let u 2 H 1 be some solution to the Poisson equation (2.2.1) and uh 2 Wh be some nite element solution satisfying (both without regard to the boundary conditions)

h : (gradh uh gradh v) = (K v) ; (gradh v F ) for all v 2 W Using integration by parts we obtain (gradh(u ; uh ) gradh v) =

X Z

T 2Th

h : (2.4.1) ( @u ; F  nT )v for all v 2 W @T @n

The interior error analysis only depends on the above interior discretization equation.

Lemma 2.4.1. Let L be a linear functional on Wh and assume that u 2 H 1 + Wh

satises

;

gradh u gradh v = L(v) for all v 2 V h :


(2.4.2)

Then for any concentric disks G0 b G b  and any nonnegative integer t

kuk0 G  C ; hk gradh uk0 G + kuk;t G + kLkG
: 0

(2.4.3)

Moreover, if L(v) = 0 for all v 2 V h , then

kuk0 G  C ; hk gradh uk0 G + kuk;t G
: 0

(2.4.4)

Proof. We rst prove that for any integer s  0,

kuk;s G  C ; hk gradh uk0 G + kuk;s;1 G + kLkG
0

(2.4.5)

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holds for any concentric disks G0 b G b  (not necessarily the same sets as in (2.4.3)). Then inequality (2.4.3) can be obtained by iteration. Find a union of elements Gh , such that G0 b Gh b G. Construct a cut-o function ! 2 C01(Gh ) such that ! = 1 on G0 . By denition ;
!u  kuk;s G0  k!uk;s G = sups kks G : (2.4.6)  2H (G) 6=0

By Lemma 2.2.1, there exists a unique function U 2 H s+2(G) \ H 1 (G), such that

;U = 

in G

U = 0 on @G: Moreover,

kU ks+2 G  C kks G:

(2.4.7)

For convenience, we extend U by zero outside the disk G. Now, we can estimate the numerator of the right hand side of (2.4.6): ;

!u 

G = ; !u  U Gh




;

= =




X ;

T 2Gh

X ;

T 2Gh

grad(!u) grad U




T

;

grad u grad(!U )
T + ; u grad ! grad U
T Z

+ u div(U grad !) T ; @T ;

 @U !u @n ds @T

Z




;



@U
ds uU @! + !u @n @n

(2.4.8)

where we use the denition of U , dierentiation rules, and integration by parts. Since supp !U  Gh  G, the continuous piecewise linear interpolant (!U )I of !U belongs to 2 V h(G), thus

k!U ; (!U )I k1 G  ChkU k2 G:

(2.4.9)

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So we have (!u ) =

X ;

T 2Gh

+

;

grad u grad(!U ; (!U )I

X ;

T 2Gh

u grad ! grad U







;

I T + L (!U )

;





T + u div(U grad ! ) T

X Z



Z @! uU @n ds + !u @U @n ds @T @T

T 2Gh

;




=: A + L (!U )I + B + C:

(2.4.10)

;




;




where we use the fact that gradh u grad(!U )I = L (!U )I . Applying (2.4.9), Lemma 2.3.2, and the Schwarz inequality, we get jAj  Chk gradh uk0 GhkU k2 G

jBj  C kuk;s;1 GkU ks+2 G jC j  Chk gradh uk0 G kU k2 G jL; (!U )I
j  C kLkG kU k1 G:

(2.4.11)

h

h

Substituting (2.4.11) into (2.4.10), then using (2.4.6) and (2.4.7) we obtain (2.4.5). To prove (2.4.3), take a family of concentric disks: G0 b G1 b : : : Gt = G. Then applying (2.4.5) with s = 0 and G replaced by G1 , we obtain

kuk0 G  C ; hk gradh uk0 G + kuk;1 G + kLkG




kuk0 G  C ; hk gradh uk0 G + kuk;2 G + kLkG




0

1

1

1

:

To bound kuk;1 G1 , we apply (2.4.5) with G0 and G replaced by G1 and G2 , respectively, and s = 1. Thus, we get 0

2

2

2

:

Continuing in this fashion, we obtain the (2.4.3). If L(v) = 0 for all v 2 V h , we see easily from the above proof that the term kLkG can be taken away from the right hand side of (2.4.3).

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2.5 The Main Result In this section, we prove the main result of this chapter: Theorem 2.5.2. To be specic, we rst use a local energy estimate to study the discrete function satisfying (2.5.1). This equation is usually satised by the dierence between the global nite element solution and the local projection of the exact solution. Then we combine it with Lemma 2.4.1 to obtain a local version of Theorem 2.5.2. The nal result is obtained by a covering argument.

Lemma 2.5.1. Let L be a linear functional on Wh and assume that u 2 Wh satises

h : (gradh u gradh v) = L(v) for all v 2 W

(2.5.1)

Then for any concentric disk G0 b G and nonnegative integer t, the following holds

kukh1 G  C ; kuk;t G + kLkG
:

(2.5.2)

0

Proof. Let G0 b G1 b G be concentric disks and Gh, G1 b Gh b G, be a union of triangles. Construct a cut-o function ! 2 C01(G1 ) such that ! = 1 on G0 . Then,

k gradh uk  k! gradh uk 2 0 G0

=

Z

G

2 = 0G

gradh u  gradh

(! 2 u)



;

Z

G

!2 gradh u  gradh u

Z

2

G



! gradh u  u grad ! = J1 + J2: (2.5.3)

Using the inverse inequality (cf. 16])

hk gradh uk0 Gh  C kuk0 Gh

(2.5.4)

the Schwarz inequality, (2.3.1), and the arithmetic-geometric mean inequality, we

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get Z

j J1 j = j G gradh u  gradh; !2u ; (!2u)I
+ L; (!2u)I
j  C k gradh uk0 G k gradh ; !2u ; (!2u)I
kG + kLkG k gradh(!2u)I k0 G  Chk gradh uk0 G ; k! gradh uk0 G + kuk0 G
; ;

+ kLkG k gradh (!2u)k0 G + k gradh !2u ; (!2 u)I k0 G  41 k! gradh uk20 G + C kuk20 G + C kLk2G: (2.5.5) h

h

h

1

1

1

1

1

1

1

1

The estimate on jJ2j is straightforward:

j J2 j  C k! gradh uk0 G kuk0 G  14 k! gradh uk20 G + C kuk20 G : 1

1

(2.5.6)

1

Combining (2.5.3), (2.5.5), and (2.5.6), then taking the square root, we obtain

k gradh uk0 G  C ; kuk0 G + kLkG
: 0

1

From (2.5.1) and Lemma 2.4.1, we have

kuk0 G  C (hk gradh uk0 G + kuk;t G + kLkG): 0

Combing the above two inequalities we get

kukh1 G  C ; kuk0 G

+ hkukh1 G + kuk;t G + kLkG :


1

0

Then using Lemma 2.4.1 again with G0 replaced by G1 to bound kuk0 G1 on the right hand side of the above inequality yields

kukh1 G  C ; hkukh1 G + kuk;t G + kLkG
: 0

We will now use an iteration method 33] to prove (2.5.2).

(2.5.7)

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Let G0 b G1 b : : : b Gt+2 = G be concentric disks and apply (2.5.7) to each pair Gj b Gj+1 (with G0 and G replaced by Gj and Gj+1 , respectively) to get

kuk11 G  C ; hkukh1 G

j+1 + kuk;t Gj+1 + kLkGj+1 :

j




Combining inequalities of above type (for j = 0 1 : : : ) we obtain

kukh1 G  C ; hkukh1 G + kuk;t G + kLkG
 :::  C ; ht+1kukh1 G + kuk;t G + kLkG 0

1

1

t+1

1

t+1

t+1




:

By (2.3.2), there is a set Gh , Gt+1 b Gh b Gt+2 = G, such that

ht+1kukh1 Gt+1  ht+1kukh1 Gh  C kuk;t Gh  C kuk;t G: Thus the above two inequalities imply (2.5.2).

2 L2 and div F 2 L2 . Suppose that u 2 H 1 satises uj 2 H 2 (1 ) and

Theorem 2.5.2. Let 0 b 1 b  and assume that K

Assume that F j1 2 H 1(1 ). 1 uh 2 Wh satises (2.4.1). Let t be a nonnegative integer. Then there exists a constant C depending only on 0 , 0, and t, and a positive number h1, such that for h 2 (0 h1 ]

ku ; uhkh1   C ; hkuk2  ku ; uhk0   C ; h2kuk2  0

0

1 1

+ hkF k1 1 + ku ; uhk;t 1




+ h2kF k1 1 + ku ; uh k;t 1 :


(2.5.8) (2.5.9)

Proof. We rst prove a local version of (2.5.8), that is,

ku ; uh kh1 G  C h; kuk2 G + kF k1 G
+ C ku ; uh k;t G 0

(2.5.10)

21

for any pair of concentric disks G0 b G b . In order to do so, we nd a disk G1 such that G0 b G1 b Gh b G, with Gh a union of triangles. Construct a cut-o function ! 2 C01(Gh ) such that ! = 1 on G1 . Use the notation ue = !u and dene

ue 2 W h(Gh ) by (gradh ue gradh v) = (gradh ue gradh v) for all v 2 W h (Gh ):

(2.5.11)

This problem is uniquely solvable. Moreover,

kue ; uekh1 G  C v2W inf(G ) kue ; vk1 G  C hkuk2 G : h

h

h

h

h

By the triangle inequality,

ku ; uhkh1 G  kue ; uekh1 G + k ue ; uhkh1 G  C hkuk2 G + k ue ; uhkh1 G : h

0

h

0

0

(2.5.12)

From (2.5.11), (2.4.1), and the fact that ! = 1 on G1, (gradh ( ue ; uh) gradh v) = (gradh (u ; uh) gradh v) X Z @u ; F  n )v = ( @n T T 2Gh @T

h(G1 ): =: L(v) for all v 2 W

By Lemma 2.3.2,

jL(v)G j  Ch(kuk2 G + kF k1 G )k gradh vk0 G h

which implies

h

h

kLkG  C h; kuk2 G + kF k1 G h

h (Gh ) for all v 2 W

h

h


h

:

(2.5.13)

22

Then, applying Lemma 2.5.1 with G replaced by G1, we obtain

k ue ; uhkh1 G  C ; k ue ; uhk;t G + kLkG )  C (k ue ; uek;t G + ku ; uhk;t G + kLkG )  C h(kF k1 G + kuk2 G
+ C ku ; uhk;t G: 1

0

h

1

1

h

(2.5.14)

By the triangle inequality, (2.5.12), and (2.5.14), inequality (2.5.10) is obtained. In order to prove (2.5.8), we use a covering argument. Let d = d0=2 where d0 = dist(  0 @ 1 ). Cover  0 with a nite number of disks G0 (xi ) i = 1 2 : : : m centered at xi 2  0 with diam G0(xi ) = d. Let G(xi ) i = 1 2 : : : k be corresponding concentric disks with diam G(xi ) = 2d. Applying (2.5.10) to each pair G0(xi ) and G(xi ), and adding inequalities of the form (2.5.10), we obtain the desired result. To prove (2.5.9), note that ;

gradh ( ue ; uh) grad v = 0 for all v 2 V h (G1):


By Lemma 2.4.1, we obtain

ku ; uhk0 G  C ; hk gradh (u ; uh)k0 G + ku ; uh k;t G 0

1


1

for any disks G0 b G1. Then, applying (2.5.10) with G0 replaced by G1 to get

ku ; uhk0 G  C ; h2kuk2 G + h2kF k1 G + ku ; uhk;t G
0

for any pair of disks G0 b G b . Then a covering argument leads to (2.5.9)

23

CHAPTER 3 INTERIOR ESTIMATES FOR THE STOKES EQUATIONS 3.1 Introduction In this chapter we establish interior error estimates for nite element approximations to solutions of the Stokes equations. The theory (cf. 6]) to be developed here covers a wide range of nite element methods for the Stokes equations. It is based on some abstract hypotheses that apply to most stable elements. This is dierent than what we did in Chapter 2, where we only studied one special element. The conclusion we obtain here is quite similar to that for the second order elliptic equation. Namely, we prove that, the approximation error of the nite element method in the interior region is bounded above by two terms: the rst one measures the local approximability of the exact solution by the nite element space and the second one, given in an arbitrary weak Sobolev norm over a slightly larger domain, represents a global pollution eect. The technique used here is adapted from that for the second order elliptic equation by Nitche and Schatz 33]. Although the general approach is not new, there are a number of signicant diculties which arise for the Stokes system that are not present in previous works. The method developed here will also be generalized to get the interior error estimate of the Arnold-Falk element for the Reissner-Mindlin plate model in the next chapter. After the preliminaries of the next section, we set out the hypotheses for the nite element spaces in section 3.3. These assumptions are satised by most stable

24

elements on a locally quasi-uniform mesh. In section 3.4, we introduce the local equations and derive some basic properties of their solutions. Section 3.5 gives the precise statement of our main result and its proof. In section 3.6, we apply the general theory to the MINI element of Arnold-Brezzi-Fortin 2] and show that it achieves the optimal convergence rate in the energy norm away from the boundary for a nonconvex polygonal domain. However this optimal convergence cannot be obtained on the whole domain due to the corner singularity of the exact solution.

3.2 Notations and Preliminaries Let  denote a bounded domain in R2 and @  its boundary. We dene the gradient of a vector function: 



@ 1 =@y : grad = @@ 12 =@x =@x @ 2 =@y Let G be an open subset of  and s an integer. If 2 H s(G),
2 H ;s(G), and ! 2 C01(G), then j(!
)j  C k ks Gk
k;s G with the constant C depending only on G, !, and s. For  2 H s (G),  2 H ;s+1(G) dene ;

R(!  ) = (grad !)t grad Then




; ; grad (grad !)t
:

jR; !  
j  C kks Gkk;s+1 G:

If, moreover,  2 H ;s+2, we have the identity ;

grad(!) grad 
= ; grad grad(!)
+ R(!  ):

(3.2.1) (3.2.2)

25

If X is any subspace of L2, then X^ denotes the subspace of elements with average value zero. The following lemma states the well-posedness and regularity of the Dirichlet problem for the generalized Stokes equations on smooth domains. (Because we are interested in local estimates we really only need this results when the domain is a disk.) For the proof see 40, Chapter I, x 2].

Lemma 3.2.1. Let G be a smoothly bounded plane domain and m a nonnegative integer. Then for any given functions F exist uniquely determined functions

2 H m;1(G), K 2 H m (G) \ L^ 2(G), there

2 H m+1(G) \ H 1(G) p 2 H m (G) \ L^2 (G) such that

grad grad

; ;div
p
= ;F

for all
2 H 1(G)

;

;

Moreover,




;

div q = K q




for all q 2 L^ 2(G):

k km+1 G + kpkm G  C ;kF km;1 G + kK km G


where the constant C is independent of F and K .

3.3 Finite Element Spaces In this section we collect assumptions on the mixed nite element spaces. As usual for the interior estimate, we require the superapproximation property of the nite element spaces, in addition to the the approximation and stability properties. Let   R2 be the bounded open set on which we solve the Stokes equations. We denote by V h the nite element subspace of H 1, and by Wh the nite element

26

subspace of L2 . For 0  , dene V h (0) = f

j j 2 V h g V h (0) = f 2 V h j supp   0g 0

Wh (0) = f pj0 j p 2 Wh g

h (0) = f p 2 Wh j supp p   0 g: W

Let G0 and G be concentric open disks with G0 b G b . We assume that there exists a positive real number h0 and positive integers k1 and k2, such that for ;  h 2 0 h0 , the following properties hold. A1. Approximation property.

(1) If 2 H m (G) for some positive integer m, then there exists a I 2 V h such that

k ; I k1 G  Chr ;1 j jm G 1

r1 = min(k1 + 1 m):

(2) If p 2 H l (G) for some nonnegative integer l, then there exists a pI such that

kp ; pI k0 G  Chr kpkl G 2

2 Wh

r2 = min(k2 + 1 l):

Furthermore, if and p vanish on G n G 0 , respectively, then I and pI can be chosen to vanish on  n G . A2. Superapproximation property. Let ! 2 C01(G), 2 V h , and p 2 Wh . Then

h(G), such that there exist
2 V h (G) and q 2 W

k! ;
k1   Chk k1 G k!p ; qk0   Chkpk0 G where C depends only on G and !.

27

A3. Inverse property. For each h 2 (0 h0], there exists a set Gh , G0 b Gh b G, such that for each nonnegative integer m there is a constant C for which

k k1 G  Ch;1;m k k;m G for all 2 V h kpk0 G  Ch;m kpk;m G for all p 2 Wh: h

h

h

h

A4. Stability property. There is a positive constant  , such that for all h 2 (0 h0] there is a domain Gh , G0 b Gh b G for which inf

;

sup

p2W^h (Gh ) 2V h (Gh ) p6=0 6 0 =

k




p Gh k1 Gh kpk0 Gh

div



When Gh = , property A4 is the standard stability condition for Stokes elements. It will usually hold as long as Gh is chosen to be a union of elements. The standard stability theory for mixed methods then gives us the following result.

Lemma 3.3.1. Let Gh be a subdomain for which the stability inequality in A4

1(Gh ) and p 2 L2(Gh ), there exist unique

holds. Then for 2 H R R and p 2 Wh (Gh ) with Gh p = Gh p such that ;

grad(

; ) grad

; ;div
p ; p
= 0 ;
div( ; ) q = 0

2 V h (Gh )

for all
2 V h (Gh ) for all q 2 Wh (Gh ):

Moreover,

k ; k1 G + kp ; pk0 G  C ;
2Vinf(G ) k ;
k1 G h

h

h

h

h

+


k p ; qk0 G : q2W (G )

inf h

h

h

The approximation properties A1 are typical of nite element spaces V h and Wh constructed from polynomials of degrees at least k1 and k2 , respectively. (It does not matter that the subdomain G is not a union of elements since and p

28

can be extended beyond G.) The inverse inequality was proved in section 2.3 for general nite element spaces. The superapproximation property is discussed as Assumptions 7.1 and 9.1 in 41]. Many nite element spaces are known to have the superapproximation property. In particular, it was veried in 33] for Lagrange and Hermite elements. To end this section we shall verify the superapproximation for the MINI element. Let bT denote the cubic bubble on the triangle T , so on T , bT is the cubic polyR nomial satisfying bT j@T = 0 and T bT = 1. We extend bT outside T by zero. For a given triangulation Th let Vh denote the span of the continuous piecewise linear functions and the bubble functions bT , T 2 Th . The MINI element uses Vh Vh as the nite element space for velocities. We wish to show that if  2 Vh and ! 2 C01(G) then k! ; k1 G  Chkk1 G for some 2 V h (G). We begin by writing  = l + b P with l piecewise linear and b = T 2Th T bT for some T 2 R. We know that there exists a piecewise linear function l supported in G for which

k!l ; l k1   Chklk1 G: P Turning to the bubble function term b dene b = T G(T LT !)bT 2 V h (G) where LT ! 2 R is the average value of ! on T . Now if T intersects supp ! then T  G, at least for h suciently small. Hence

k!b ; bk20  = X k!b ; bk20 T = X kT bT (! ;LT !)k20 T T G

T G

 X k! ;LT !k2L1(T )kT bT k20 T  Ch2kbk20 G T G

29

where the constant C depends on !. Moreover,

k grad(!b ; b)k20  = X k grad(!b ; b)k20 T = =

X

T G X

T G

T G

k grad; T bT (! ;LT !)
k20 T kT (w ;LT !) grad bT + T bT grad(! ; LT !)k20 T

 C ; h2 X k grad !k21 T kT grad bT k20 T + k grad !k21 T X kT bT k20 T
T G

T G

 Ch2k bk21 G where we used the fact that

kbT k0 T  C hkbT k1 T : Taking h = b + l 2 V h (G) we thus have

k!h ; h k1   C h; kbk1 G + klk1 G
: We complete the proof by showing that kbk1 T + klk1 T  C kb + lk1 T for any triangle T with the constant C depending only on the minimum angle of T . Since R T grad b  grad l = 0, it suces to prove that

kbk0 T + klk0 T  C kb + lk0 T : If T is the unit triangle this hold by equivalence of all norms on the nite dimensional space of cubic polynomials, and the extension to an arbitrary triangle is accomplished by scaling.

30

3.4 Interior Duality Estimates Let ( p) 2 H 1 L2 be some solution to the generalized Stokes equations

; + grad p = F div = K: Regardless of the boundary conditions used to specify the particular solution, ( p) satises ;

grad grad

; ; div
p
= ; F

for all
2 H 1 ;




;

div q = K q

for all q 2 L2:




Similarly, regardless of the particular boundary conditions, the nite element solution ( h ph ) 2 V h Wh satises ;

grad

h

grad

; ; div
ph
= ; F

for all
2 V h ;

div h q = K q




;







; ; div
p ; ph
= 0

h : for all q 2 W

Therefore ;

grad(

;

h)

grad


;

div(

;

h)

for all
2 V h

q = 0 for all q 2 Wh :

(3.4.1)




(3.4.2)

The interior interior error analysis starts from these interior discretization equations.

Theorem 3.4.1. Let G0 b G be concentric open disks with closures contained in  and s an arbitrary nonnegative integer. Then there exists a constant C such that

31

if ( p) 2 H 1 L2, and ( h ph ) 2 V h Wh satisfy (3.4.1) and (3.4.2), we have

k ; hk0 G + kp ; ph k;1 G  C ; hk ; hk1 G + hkp ; phk0 G
+ k ; h k;s G + kp ; ph k;1;s G : 0

0

(3.4.3)

In order to prove the theorem we rst establish two lemmas.

Lemma 3.4.2. Under the hypotheses of Theorem 3.4.1, there exists a constant C for which

kp ; phk;s;1 G  C ; hk ; h k1 G + hkp ; phk0 G
+ k ; h k;s;1 G + kp ; ph k;s;2 G : 0

Proof. Choose a function ! 2 C01(G) which is identically 1 on G0 . Also choose a function 2 C01(G0 ) with integral 1. Then

kp ; ph k;s;1 G  k!(p ; ph)k;s;1 G = 0

Now

;

and clearly

!(p ; ph ) g = !(p ; ph ) g ;


;

;

sup

g2H s+1(G) g6=0

Z

G

!(p ; ph ) g kgks+1 G :


g + !(p ; ph )


;




Z

G

(3.4.4)

g

Z

j !(p ; ph) G g j C kp ; ph k;s;2 Gkgk0 G: R Since g ; G g 2 H s+1(G) \ L^ 2 (G) it follows from Lemma 3.2.1 that there exist  2 H s+2(G) \ H 1 (G) and P 2 H s+1 (G) \ L^ 2(G) such that ;

;

grad grad








; ; div
P
= 0 ;

1(G) for all
2 H

div  q = g ;


;

Z

G

gq




(3.4.5)

for all q 2 L2(G): (3.4.6)

32

Furthermore,

kks+2 G + kP ks+1 G  C kgks+1 G: (3.4.7) Then, taking q = !(p ; ph) in (3.4.6), we obtain Z ;
g ; g !(p ; ph ) G ;
;
;
= div  !(p ; ph ) = div(!) p ; ph ; grad ! (p ; ph) ;
n;
;
o = div(!)I p ; ph + div! ; (!)I ] p ; ph ; grad ! (p ; ph ) =: A1 + B1:

(3.4.8)

Here the superscript I is the approximation operator specied in property A1 of section 3.3. Choosing
= (!)I in (3.4.1), we get

A1 := div(!)I p ; ph = grad( ;




;

;

grad(!)I
;
;
= grad( ; h ) grad(!) + grad( ; h ) grad(!)I ; (!)] ;
n ;
= grad!( ; h )] grad + R !  ; h ;
o + grad( ; h ) grad(!)I ; !] =: A2 + B2 (3.4.9) h)

where R is dened in (3.2.1). Next, setting
= !( ;

A2 := grad!( ;

= div( ;

= div(

; ;

;

;

h ) in (3.4.5), we obtain

grad
= ; div!( ; h )] P

;
h ) !P + grad ! P ( ; h )
;
h ) !P ; (!P )I + grad ! P ( ; h ) h )]

where we applied (3.4.2) in the last step. Applying the approximation property A1 and (3.2.2) we get

jB1j  C ; hkk2 Gkp ; ph k0 G + kks+2 Gkp ; ph k;s;2 G
jB2j  C ; k ; h k;s;1 Gkks+2 G + hk ; h k1 Gkk2 G
jA2j  C ; hk ; hk1 GkP k1 G + k ; h k;s;1kP ks+1 G
:

(3.4.10)

33

Substituting (3.4.7) into (3.4.10) and combining the result with (3.4.4), (3.4.8), and (3.4.9), we arrive at (3.4.3). Now we state the second lemma to be used in the proof of Theorem 3.4.1.

Lemma 3.4.3. Under the hypotheses of Theorem 3.4.1, there exists a constant C for which

k ; hk;s G  C ; hk ; h k1 G + hkp ; ph k0 G
+ k ; h k;s;1 G + kp ; ph k;s;2 G : 0

Proof. Given F by ;

2 H s(G), dene  2 H s+2(G) \ H 1(G) and P 2 H s+1(G) \ L^2 (G)

grad grad

; ; div
P
= ; F

for all
2 H 1(G) (3.4.13) ;
div  q = 0 for all q 2 L2 (G): (3.4.14)

Then, by Lemma 3.2.1,

kks+2 G + kP ks+1 G  C kF ks G Now

k ; hk;s G  k!( ; h )k;s G = 0

C = C (G0 G): sup

;

!(

 s (G) F 2H F 6=0

with ! as in the proof of the previous lemma. Setting
= ! we get ;

!(

;




; )F
kF ks G

;

;




h in (3.4.13),

grad grad!( ; h )]
; ; div!( ; h )] P
n;
;
o = grad(!) grad( ; h ) ; div( ; h ) !P n o ; R; !  ; h
+ ; grad ! P ( ; h )
=: E1 + F1

h) F =

;

34

To estimate E1, we set q = (!P )I in (3.4.2) and obtain ;

E1 = grad(!)I grad(

; grad! ; ;

;

(!)I ]

h)




;

grad(

n;

;

div( h)


o

;

h)

!P ; (!P )I




=: E2 + F2 :

Taking
= (!)I in (3.4.1), we arrive at ;

E2 = grad(!)I grad( = div(!)I p ; ph ;




;

h)




= div(!) p ; ph + div(!)I ; (!)] p ; ph ;




;




= grad ! (p ; ph ) + div(!)I ; (!)] p ; ph ;




;




where we applied (3.4.14) in the last step. Applying (3.2.2) and the approximation property A1, we have

j F1 j C (k ; h k;s;1 Gkks+2 G + k ; hk;s;1 GkP ks+1 G) j F2 j  Ch(k ; h k1 GkP k1 G + k ; h k1 Gkk2 G) j E2 j  C ; kp ; phk;s;2 Gkks+2 G + h kp ; ph k0 Gkk2 G
: From these bounds we get the desired result. Proof of Theorem 3.4.1. Let G0 b G1 b : : : Gs = G be concentric disks. First applying Lemma 3.4.2 and Lemma 3.4.3 with s replaced by 0 and G replaced by G1 , we obtain

k ; h k0 G + kp ; ph k;1 G  C ; hk ; h k1 G + hkp ; phk0 G
+ k ; h k;1 G + kp ; ph k;2 G : To estimate k ; h k;1 G and kp ; phk;2 G , we again apply Lemma 3.4.2 and 0

0

1

1

1

1

1

1

Lemma 3.4.3, this time with G0 and G being replaced by G1 and G2 and s replaced

35

by 1. Thus, we get

k ; h k0 G + kp ; ph k;1 G  C ; hk ; h k1 G + hkp ; phk0 G
+ k ; h k;2 G + kp ; ph k;3 G : 0

0

2

2

2

2

Continuing in this fashion, we obtain (3.4.3).

3.5 Interior Error Estimates In this section we state and prove the main result of this chapter, Theorem 3.5.3. First we obtain in Lemma 3.5.1 a bound on solutions of the homogeneous discrete system. In Lemma 3.5.2 this bound is iterated to get a better bound, which is then used to establish the desired local estimate on disks. Finally Theorem 3.5.3 extends this estimate to arbitrary interior domains.

Lemma 3.5.1. Suppose ;

grad

h

;

h ph

grad








2 V h Wh satises

; ; div
ph
= 0

for all
2 V h ;


h : div h q = 0 for all q 2 W

(3.5.1) (3.5.2)

Then for any concentric disks G0 b G b , and any nonnegative integer t, we have

k hk1 G + kph k0 G  C ; hk h k1 G + hkphk0 G + k hk;t G + kphk;t;1 G
0

0

(3.5.3)

where C = C (t G0 G). Proof. Let Gh, G0 b Gh b G, be as in Assumption A4. Let G0 be another disk concentric with G0 and G, such that G0 b G0 b Gh , and construct ! 2 C01(G0 )

1 (G0 ), f with ! 1 on G0. Set fh = ! h 2 H ph = !ph 2 L2 (G0 ). By Lemma 3.3.1,

36

we may dene functions fh 2 V h (Gh ) and f ph 2 Wh (Gh ) by the equations ;


grad( fh ; fh ) grad

; ; div
pfh ; f ph = 0 for all
2 V h(Gh )

(3.5.4)

;

div( fh ; fh ) q = 0 for all q 2 Wh(Gh ) (3.5.5)


together with Gh ( f ph ; f ph) = 0. Furthermore, there exists a constant C such that R

k fh ; fh k1 G + kpfh ; fphk0 G  C ;
2Vinf(G ) k fh ;
k1 G + q2Winf(G ) kpfh ; qk0 G  Ch(k hk1 G0 + kph k0 G0 ) h

h

h

h

h

h


h

h

(3.5.6)

where we have used the superapproximation property in the last step. To prove (3.5.3), note that

k hk1 G + kphk0 G  k fh k1 G + kpfhk0 G  k fh ; fh k1 G + kpfh ; fph k0 G + k fh k1 G + k fphk0 G  Ch(k hk1 G0 + kph k0 G0 ) + k fhk1 G + k fphk0 G : (3.5.7) Next, we bound k fh k1 G . In (3.5.4) we take
= fh to obtain, for a positive 0

h

0

h

h

h

h

h

h

h

h

constant c,

ck fh k21 Gh  grad fh grad fh ; ;

= grad fh grad fh







; ; div fh fph ; fph
:

(3.5.8)

For the rst term on the right hand side of (3.5.8), we have ;




;




grad fh grad fh = grad(! h ) grad fh ;
;
= grad h grad(! fh ) ; R ! fh h ;
n;
= grad h grad(! fh )I + grad h grad! fh ; (! fh )I ]

;R; ! fh

h


o

=: G1 + H1 :

(3.5.9)

37

To bound G1, we take
= (! fh )I in (3.5.1) and get ;

G1 = div(! fh )I ph




= div(! fh ) ph + div(! fh )I ; ! fh ] ph ;




;




= div fh !ph + grad ! ph fh + div(! fh )I ; ! fh ] ph


;




;

;




= div fh pfh + grad ! ph fh + div(! fh )I ; ! fh ] ph




=: div fh pfh + H2 :

(3.5.10)

;




;

;




;




Combining (3.5.7), (3.5.8), (3.5.9), and (3.5.10), we obtain

ck fh k21 Gh  div fh pfh + H1 + H2 ; div fh pfh ; f ph ;




;

;




= div fh f ph + H1 + H2 :




(3.5.11)

Taking q = f ph in (3.5.5), we get ;




;




;

div fh f ph = div fh f ph = div(! h ) f ph ;




;




= div h ! f ph + grad ! f ph h




= div h ! f ph ; (! f ph )I + grad ! f ph h =: H3 (3.5.12) ;




;




where we used (3.5.2) at the last step. Applying the Schwartz inequality, (3.2.2), and the superapproximation property A2, we get

j H1 j C ; h k h k1 G0 + k h k0 G0
k fh k1 G j H2 j C ; kph k;1 G0 + h kph k0 G0
k fhk1 G j H3 j C ; h k h k1 G0 + k h k0 G0
k fph k0 G : h

h

h

Combining the above three inequalities with (3.5.11) and (3.5.12), and using the arithmetic-geometry mean inequality, we arrive at

k fhk21 G  C1; h2k h k21 G0 + k h k20 G0 + h2kphk20 G0 + kphk2;1 G0
;
+ C2 k h k0 G0 + h k h k1 G0 k f ph k0 G : (3.5.13) h

h

38

Next we estimate k f ph k0 Gh . By the triangle inequality,

k fphk0 G  h

   ph  f 



R

Z 

f ph   G h ; 1 ; meas(Gh )  + meas(Gh )  G ( fph ; fph) h 0 Gh 0 Gh

Z  + meas(Gh );1 

Gh

  ph  f

0 Gh

:

(3.5.14)

Notice that the second term on the right hand side of (3.5.14) is bounded above by the right hand side of (3.5.6), and, for the last term, Z   

Gh

  ph f

0 Gh

=

Z   

Gh

  !ph 

0 Gh

 C kphk;1 G0 :

(3.5.15)

To estimate the rst term, we use the inf-sup condition,    ph  f 



R

;

div
f ph ph  Gh f  C sup ; meas(  Gh ) 0 Gh k
k1 Gh
2V h (Gh )




Gh :

(3.5.16)


6=0

To deal with the numerator on the right hand side of (3.5.16), we apply (3.5.4), ;




; ; grad( fh ; fh ) grad

;
;
= div
!ph ; grad( fh ; fh ) grad
;
;
;
= div(!
) ph ; grad( fh ; fh ) grad
; grad ! ph
;
;
= div(!
)I ph ; grad( fh ; fh ) grad
;
;
+ div(!
; (!
)I ) ph ; grad ! ph
: (3.5.17) ;

div
f ph = div
pfh




;




We use (3.5.1) to attack div(!
)I ph and get ;




;

div(!
)I ph = grad h grad(!
)I




= grad h grad(!
) + grad h grad(!
)I ; !
] ;




;




= grad(! h ) grad
+ R !
h + grad h grad(!
)I ; !
] ;

;







;

=: grad fh grad
+ M1 :




;




(3.5.18)

39

Combining (3.5.17) and (3.5.18), we get ;




div
f ph =

;

div(w
; (w
)I ) ph

;







; ; grad w ph



+ grad fh grad
+ M1 ;




=: grad fh grad
+ M1 + M2:

(3.5.19)

Then applying the superapproximation property, the Schwartz inequality, and (3.2.2), we arrive at j M1 j C ; k h k0 G0 + h k hk1 G0
k
k1 Gh

j M2 j C ; h kph k0 G0 + kphk;1 G0
k
k1 G j ; grad fh grad

j k fh k1 G k
k1 G : h

h

h

Combining (3.5.14), (3.5.15) , (3.5.16), and (3.5.19) with the above three inequalities, we obtain

k fphk0 G  C ; h k h k1 G0 + k h k0 G0 + h kphk0 G0 + kph k;1 G0 + k fh k1 G




: (3.5.20)

h

h

Substituting (3.5.20) into (3.5.13), we obtain

k fh k1 G  C ; h k hk1 G0 + k h k0 G0 + h kph k0 G0 + kphk;1 G0
: (3.5.21) Thus, substituting (3.5.21) back into (3.5.20), we nd that k f ph k0 G is also bounded h

h

above by the right hand side of (3.5.21). Therefore, from (3.5.7) we obtain

k hk1 G + kphk0 G  C ; h k h k1 G0 + k h k0 G0 + h kph k0 G0 + kphk;1 G0
: 0

0

Applying Theorem 3.4.1 for the case that = p = 0 and G0 in place of G0 , we nally arrive at

k h k1 G + kphk0 G  C ; h k h k1 G + k h k;t G + h kphk0 G + kph k;t;1 G
: 0

0

40

Lemma 3.5.2. Suppose the conditions of Lemma 3.5.1 are satised. Then

k hk1 G + kph k0 G  C ; k h k;t G + kph k;t;1 G
: 0

(3.5.22)

0

Proof. Let G0 b G1 b ::: b Gt+2 = G be concentric disks and apply Lemma 3.5.1 to each pair Gj b Gj+1 to get

k hk1 G + kphk0 G  C ; hk hk1 G + hkphk0 G j

j

j+1

+ k h k;t Gj+1 + kph k;t;1 Gj+1 :


j+1

(3.5.23)

Combining these we obtain

k hk1 G + kphk0 G  C ; ht+1k hk1 G + ht+1kph k0 G
+ k h k;t G + kph k;t+1 G : 0

t+1

0

t+1

t+1

t+1

(3.5.24)

While by A3, we can nd Gh , Gt+1 b Gh b Gt+2 = G, such that

ht+1k

k  ht+1k h k1 G  C k h k;t G  C k h k;t G ht+1kph k0 G  ht+1kph k0 G  C kph k;t;1 G  C kph k;t;1 G : h 1 Gt+1

h

t+1

h

h

(3.5.25)

h

Thus inequality (3.5.22) follows from (3.5.23), (3.5.24), and (3.5.25). We now state the main result of the chapter.

Theorem 3.5.3. Let 0 b 1 b  and suppose that ( p) 2 H 1 L2 (the exact

solution) satises j1 2 H m (1 ) and pj1 2 H m;1 (1 ) for some integer m > 0. Suppose that ( h ph ) 2 V h Wh (the nite element solution) is given so that (3.4.1) and (3.4.2) hold. Let t be a nonnegative integer. Then there exists a constant C depending only on 1, 0, and t, such that

k ; hks  + kp ; phks;1   C ; hr ;s k km  + hr ;skpkm;1 
+ k ; h k;t  + kp ; ph k;t;1  0

0

1

1

1

2

1

1

s=0 1 (3.5.26)

41

with r1 = min(k1 + 1 m), r2 = min(k2 + 2 m), and k1, k2 as in A1.

The theorem will follow easily from a slightly more localized version.

Theorem 3.5.4. Suppose the hypotheses of Theorem 3.5.3 are fullled and, in addition, that 0 = G0 and 1 = G1 are concentric disks. Then the conclusion of the theorem holds. Proof. Let G00 b G0 be further concentric disks strictly contained between G0 and G and let Gh be a domain strictly contained between G0 and G for which properties A3 and A4 hold. Thus

G0 b G00 b G0 b Gh b G b : Take ! 2 C01(G0 ) identically 1 on G00 and set e = ! , pe = !p. Let e 2 V h (G),

pe 2 Wh (G) be dened by ;

grad( e ; e) grad

;; div
pe ; pe
= 0 for all
2 V h (Gh ) ;

R

div( e ; e) q = 0 for all q 2 Wh (Gh )

(3.5.27)




(3.5.28)

R

together with Gh pe = Gh pe. Then using Lemma 3.3.1 and A1 we have

k e ; ek1 G + kpe ; pek0 G  C ;
2Vinf(G ) k e ;
k1 G + q2Winf(G ) kpe ; qk0 G  C ; hr ;1k km G + hr ;1 kpkm;1 G
: h

h

1

Let us now estimate k

h

h

h

h

; hk1 G

0

2

h

h

h


h

(3.5.29)

and kp ; phk0 G0 . First, the triangle inequality

42

gives us

k ; hk1 G + kp ; phk0 G  k ; ek1 G + kp ; pek0 G + k e ; h k1 G + k pe ; phk0 G  k e ; ek1 G + kpe ; pek0 G + k e ; h k1 G + k pe ; ph k0 G  C ;hr ;1k km G + hr ;1 kpkm;1 G
+ k e ; h k1 G + k pe ; ph k0 G : 0

0

0

0

0

0

h

h

0

0

1

h

2

h

0

0

(3.5.30)

From (3.5.27), (3.5.28) and (3.4.1), (3.4.2) we nd

; e) grad

; ; div
ph ; pe
= 0 for all
2 V h (G00) ;


h(G00 ): div( h ; e) q = 0 for all q 2 W We next apply Lemma 3.5.2 to h ; e and ph ; pe with G replaced by G00. Then ;

grad(

h

it follows from (3.5.22) that

k h ; ek1 G + kph ; pek0 G  C ; k h ; ek;t G0 + kph ; pek;t;1 G0
 C ; k ; hk;t G0 + kp ; phk;t;1 G0 + k ; ek;t G0 + kp ; pek;t;1 G0
  e e  C k ; hk;t G + kp ; ph k;t;1 G + k ; k1 G + kpe ; pek0 G : 0

0

0

0

0

0

0

h

0

h

In the light of (3.5.30), (3.5.29), and the above inequality, we have

k ; hk1 G + kp ; ph k0 G  ;hr ;1 k km G + hr ;1kpkm;1 G + k ; h k;t G + kp ; ph k;t;1 G) : 0

0

1

2

(3.5.31)

Thus, we have proved the desired result for s = 1. For s = 0, we just apply Theorem 3.4.1 to the disks G0 and G0 and get

k ; h k0 G + kp ; ph k;1 G  C (hk ; h k1 G0 + hkp ; ph k0 G0 + k ; h k;t G0 + kp ; phk;t;1 G0 ) : 0

0

43

Then, applying (3.5.31) with G0 replaced by G0 , we obtain the desired result

k ; hk0 G + kp ; phk;1 G  C ; hr k km G + hr kpkm;1 G
+ k ; h k;t G + kp ; phk;t;1 G : 0

1

0

2

Proof of Theorem 3.5.3. The argument here is same as in Theorem 5.1 of 33]. Let d = d0 =2 where d0 = dist(  0 @ 1). Cover  0 with a nite number of disks G0(xi ), i = 1 2 : : : m centered at xi 2  0 with diam G0 (xi ) = d. Let G(xi ), i = 1 2 : : : k be corresponding concentric disks with diam G(xi ) = 2d. Applying Theorem 3.5.4, we have

k ; h ks G (x ) + kp ; phks;1 G (x )  Ci ; hr ;sk km G(x ) + hr ;skpkm;1 G(x )
+ k ; h k;t G(x ) + kp ; ph k;t;1 G(x ) : 0

i

0

i

1

i

2

i

i

i

(3.5.32)

Then the inequality (3.5.26) follows by summing (3.5.32) for every i.

3.6 An Example Application As an example, we apply our general result to the Stokes system when the domain is a non-convex polygon, in which case the nite element approximation does not achieve the optimal convergence rate in the energy norm on the whole domain, due to the boundary singularity of the exact solution. Assume that  is a non-convex polygon. Then it is known that the solution of the Stokes system satises

2 H s+1 \ H01 p 2 H s 2 H 2(1 ) p 2 H 1 (1)

if 1 b 

for s < s , where s is a constant which is determined by the largest interior angle of  (cf. 19]). For a non-convex polygonal domain we have 1=2 < s < 1. The

44

value of s for various angles have been tabulated in 19]. For example, for an L-shaped domain, s  0:544. The MINI element was introduced by Arnold, Brezzi and Fortin in 2] as a stable Stokes element with few degrees of freedom. Here the velocity is approximated by the space of continuous piecewise linear functions and bubble functions and the pressure is approximated by the space of continuous piecewise linear functions only. Globally we have

k ; hk1 + kp ; ph k0  Chs;k ks+1 + kpks
which re$ects a loss of accuracy due to the singularity of the solutions. In order to apply Theorem 3.5.3, we note that a standard duality argument as in 2] gives us k ; hk0 + kp ; ph k;1  Ch2skF k0: Hence, according to Theorem 3.5.3, for 0 b 1 b , we have

k ; h k1  + kp ; phk0   C ;hk k2  0

0

1

+ hkpk1 1 + h2s kF k0 :


Since 2s > 1, the nite element approximation achieves the optimal order of convergence rate in the energy norm in interior subdomains.

45

CHAPTER 4 INTERIOR ESTIMATES FOR A FINITE ELEMENT METHOD FOR THE REISSNER-MINDLIN PLATE MODEL 4.1 Introduction The Reissner-Mindlin plate model describes deformation of a plate with small to moderate thickness subject to a transverse load. The nite element method for this model was studied extensively (cf. 11], 30], and references therein) and it has been known for a long time that a direct application of standard nite element methods usually leads to unreasonly small solution, as the plate thickness approaches zero. This is usually called the \locking" phenomenon of the nite element method for the Reissner-Mindlin plate 11], 30]. The reason behind the locking phenomenon is well known: as the plate thickness becomes very small, the numerical scheme tries to enforce a discrete version of the Kircho constraint on the displacement and the rotation ber normal to the midplane. If the nite element spaces for those two quantities are not chosen wisely, then, together with boundary conditions, the numerical solution reduces to the trivial solution. Another diculty relating to the Reissner-Mindlin plate model is that the solution possesses boundary layers, having the plate thickness as the singular parameter. As usual, the strength of the boundary layer is sensitive to the boundary condition. The structure of the dependence of the solution on the plate thickness was analyzed in detail by Arnold and Falk 4], 5].

46

The purpose of this chapter is to obtain the interior error estimate for the ArnoldFalk element 3] for the Reissner-Mindlin plate model. This element is the rst to achieve a locking-free rst order (optimal) convergence for the Reissner-Mindlin plate (under the hard clamped boundary condition). However, it does not retain the same order of convergence rate for the plate under the soft simply supported boundary condition, due to a stronger boundary layer eect. By applying the interior estimate to the soft simply supported plate, we are able to obtain the interior convergence rate of the Arnold-Falk element and show that it still possesses (almost) rst order convergence rate in the region away from the boundary. The construction of the Arnold-Falk element is based on an equivalence between the plate equations and an uncoupled system of two Poisson equations plus a Stokeslike system 3]. Arnold and Falk used the nonconforming linear element for the Poisson equation and the MINI element for the Stokes-like system. So the (global or interior) analysis of the Arnold-Falk element consists of two parts: one for the nonconforming method for the Poisson equation and another for the MINI element for the Stokes-like system. Recall that in Chapter 2 we obtained interior estimates for the nonconforming element for the Poisson equation. So the task here is essentially to analyze the interior error estimate of the MINI element for the Stokes-like system. The organization of chapter is as follows. Section 4.2 presents the ReissnerMindlin plate equations and its reformulation under the Helmholtz decomposition for the shear stress. The interior regularity of the solution of the singularly perturbed system is studied in section 4.3. The Arnold-Falk element is introduced in section 4.4. Section 4.5 is devoted to the interior duality analysis of the variant of the Stokes system. In section 4.6 we rst obtain the interior estimate of the MINI element (Theorem 4.6.2) for the Stokes-like system with perturbation and then use it to get the interior estimate of the Arnold-Falk element for the Reissner-Mindlin

47

plate model (Theorem 4.6.3), which is the main result of the chapter. As an application of the general theory we develop, we consider the soft simply supported plate in section 4.7. We will show that globally, the Arnold-Falk element only achieves (almost) h1=2 order convergence for the rotation (Theorem 4.7.3), but away from the boundary layer, (almost) optimal order convergence rate can be obtained (Theorem 4.7.4). Finally, numerical results are shown in section 4.8 which conrm the theoretical predictions.

4.2 Notations and the Reissner{Mindlin Plate Model The following operators are standard. 







11 =@x + @t12 =@y div tt1121 tt1222 = @t @t21=@x + @t22=@y   ; @p=@y curl p = @p=@x rot = @1=@y ; @2 =@x:

Let  denote the region in R2 occupied by the midsection of the plate, and denote by w and the transverse displacement of  and the rotation of the bers normal to , respectively. Under the soft simply supported boundary condition, the Reissner-Mindlin plate model determines (w ) as the unique solution to the following variational problem: Find (w ) 2 H 1 H 1 such that

a(
)+ t;2 (

;grad w
; grad ) = (g )

for all (
) 2 H 1 H 1: (4.2.1)

Here g is the scaled transverse loading function, t the plate thickness,  = E=2(1 +  ) with E the Young's modulus,  the Poisson ratio, and  the

48

shear correction factor. The bilinear form a is    Z  E @ @ @ @ @ @ 2 1 2 1 1 2 a(
) = 12(1 ;  2 ) +  +  + @y @x @x @y @y  @x    @ @ @ @ 1 ;  1 2 1 2 + 2 @y + @x @y + @x

=

Z



C E ( ) : E (
):

Here, E ( ) is the symmetric part of the gradient of and C is a fourth order tensor dened by the bilinear form a. Following Brezzi and Fortin 15], equation (4.2.1) can be reformulated by using the Helmholtz Theorem to decompose the shear strain vector

t;2(grad w ; ) = grad r + curl p

(4.2.2)

with (r p) 2 H 1 H^ 1 . Equation (4.2.1) now becomes Find (r p w) 2 H 1 H 1 H^ 1 H 1 such that (grad r grad ) = (g ) for all 2 H 1

; ; curl p

= ; grad r

for all
2 H 1 ;( curl q) ; ;1t2(curl p curl q) = 0 for all q 2 H^ 1 (grad w grad s) = ( + ;1t2 grad r grad s) for all s 2 H 1:

;

C E ( ) E (
)




(4.2.3) (4.2.4) (4.2.5) (4.2.6)

Obviously the function r in (4.2.3) is independent of t and the functions , p, and w are not. It has been shown in 5] that the transverse displacement w does not suer from the boundary layer eect under all boundary conditions. However, the regularity of solution ( p) for system (4.2.4) and (4.2.5) depends on the boundary condition imposed on the plate. For example, under the hard clamped boundary

49

1, rather than H 1 ), the following holds condition (then, is to be found in space H 3] k k2 + kpk1  C kgk0 with the constant C independent of the plate thickness t. This guarantees the MINI element to achieve a locking free rst order convergence rate for the system (4.2.4) and (4.2.5) 3]. But the above estimate does not hold for the soft simply supported plate. In this case, one can only expect that the H 3=2 norm of function and the H 1=2 norm of function p to be bounded above, independent of the small parameter t 5]. This is obviously not enough for the nite element method to achieve rst order convergence rate. It is also easy to see that a complete understanding of the dependence of the regularity of the solution on the small parameter t is of crucial importance for the convergence analysis of the nite element method. However, for the purpose of interior estimates, we need only know the inteiror regularity of the solution of the Stokes-like system. This will be given in the next section. In the following, we introduce some notations that will be used in the interior estimate. Let G be an open subset of , ! 2 C01(G), and s an integer. For  2 H s(G),  2 H ;s+2(G), P 2 H s (G), and Q 2 H ;s+2 (G), dene R(!  ) = C E (!) E () ;

and Then

;




;




; ; C E () E (!)





; ; curl P curl(!Q)
:

R0 ! P Q = curl(!P ) curl Q

jR; !  
 C kkt Gkk;t+1 G

(4.2.7)

50

and

jR0 ; ! P Q
 C kP kt GkQk;t+1 G for non-negative integers t  s.

(4.2.8)

4.3 An Interior Regularity Result In this section we present an interior regularity result for the solution of the singularly perturbed Stokes-like system under the homogeneous Dirichlet boundary condition. We will show that the regularity of the solution in the interior region is not aected by the boundary layer. This will be used in section 4.5 for the the interior duality analysis of the MINI element. The proof basically follows that in 3, Theorem 7.1] for proving the regularity of the solution of the hard-clamped plate and uses the standard approach for analyzing interior regularities for solutions of elliptic equations.

Theorem 4.3.1. Let F 2 H s (G) and K 2 H s+1(G) \ L^ 2(G), where integer s  0

1 (G) and G is a disk. Then there exists a unique solution ( P ) 2 H s+2(G) \ H H s+1(G) \ L^ 2 (G) such that

1 (G) (C E () E ()) ; (curl P ) = ( F ) for all  2 H

;( curl Q) ; ;1t2(curl Q curl P ) = (Q K )

(4.3.1)

for all Q 2 H 1 (G): (4.3.2)

Moreover,

kk2 G + kP k1 G + tkP k2 G + t2kP k3 G  C ; kF k0 G + kK k1 G
(4.3.3) kks+2 G + kP ks+1 G + tkP ks+2 G + t2kP ks+3 G  C ; kF ks G + kK ks+1 G
0

0

for an arbitrary disk G0 b G.

0

0

(4.3.4)

51

Proof. The inequality

kk2 G + kP k1 G + tkP k2 G  C ; kF k0 G + kK k1 G


is proved in 3] for K = 0 when a( ) is simplied into (grad grad), i.e., (C E () E ()) is replaced by (grad grad) in (4.3.1). By checking the proof ;


1, there and using the fact that bilinear form C E () E () is coercive on space H we can conclude that the same estimate still applies to the current case. What we will do next is to follow the same proof to show that the estimate is still true for K 6= 0. At the same time, we will prove that t2kP k3 G is also bounded above by the right hand side of (4.3.3).

1(G) L^ 2(G) as the solution of (4.3.1) and (4.3.2) with t Dene (0 P 0) 2 H set equal to zero:

1(G) (C E () E (0 )) ; (P 0 rot ) = ( F ) for all  2 H (4.3.5)

;(rot 0 Q) = (Q K )

for all Q 2 L2(G):

(4.3.6) This is a Stokes like system which admits a unique solution. Moreover, the standard regularity theory gives 40]

k0k2 G + kP 0k1 G  C ; kF k0 G + kK k1 G
:

From (4.3.1), (4.3.2), (4.3.5), and (4.3.6), we get ;

C E ( ; 0 ) E ()

;




; ; curl(P ; P 0) 
= 0

1 (G) for all  2 H

 ; 0 curl Q
+ ;1 t2; curl P curl Q
= 0 for all Q 2 H 1 (G)

which imply

; ; curl(P ; P 0) 
+ ;  ; 0 curl Q
;
+ ;1 t2 curl(P ; P 0) curl Q ;


1 (G) H 1 (G): = ;;1 t2 curl P 0 curl Q for all ( Q) 2 H

;

C E ( ; 0 ) E ()




(4.3.7)

52

Choosing  =  ; 0 and Q = P ; P 0 , we obtain

k ; 0k21 G + t2kP ; P 0k21 G  Ct2kP 0k1 GkP ; P 0k1 G: It easily follows that

k ; 0 k1 G + tkP ; P 0k1 G  CtkP 0k1 G  Ct(kF k0 G + kK k1 G): Hence also

(4.3.8)

kP k1 G  C (kF k0 G + kK k1 G):

Applying standard estimates for second-order elliptic problems to (4.3.1), we further obtain kk2 G  C (kP k1 G + kF k0 G)  C (kF k0 G + kK k1 G): (4.3.9) Now from (4.3.2) and the denition of 0 (i.e., (4.3.6)) we get

;1t2 (curl P curl Q) = ;( curl Q) ; (K Q) = (0 ;  curl Q)

for all Q 2 H 1(G):

Thus P is the weak solution of the boundary value problem

;  P = t;2 rot(0 ; )

in G

@P = 0 on @G @n

and by standard a priori estimates

kP k2 G  Ct;2k ; 0k1 G  Ct;1(kF k0 G + kK k1 )

(4.3.10)

and

kP k3 G  Ct;2k ; 0k2 G  Ct;2(kk2 G + k0k2 G)  Ct;2(kF k0 G + kK k1 G)

(4.3.11)

53

where we apply (4.3.8) in deriving (4.3.10), and (4.3.7), (4.3.9) in deriving (4.3.11). This completes the proof of (4.3.3). In order to prove (4.3.4), we take a disk G1 such that G0 b G1 b G. Find a cut-o function ! 2 C01(G1 ) with ! = 1 on G0 . We will use the notation 0 = Dx or Dy , say for example, P 0 can be either Px or Py . Then, by dierentiation rules, it is easy to obtain

; div C E (!0 ) ; curl(!P 0 ) = !F 0 ; J(! 0 ) ; P 0 curl ! =: F1

(4.3.12)

=: K1

(4.3.13)

; rot(!0) + ;1t2 (!P 0) = !K 0 ; curl !  0 + ;1 t2  !P 0 + 2;1 t2 grad !  grad P 0 where

J(! 0 ) =: div C E (!0 ) ; ! div C E (0 )

with

jJ(! 0 )j  C k0 k1 G : 1

Obviously,

Z

G1

K1 =

Z

;

G1

=;

; rot(!0 ) + ;1t2 (!P 0 )


Z

@G1

;

!0  s ; ;1 t2@ (!P 0 )=@n = 0


because both ! and grad ! vanish on @G1 . Moreover, we see that (!0 !P 0 ) satises

1 (G1) (C E () E (!0 )) ; (curl(!P 0 ) ) = ( F1 ) for all  2 H (4.3.14)

;(!0 curl Q) ; ;1t2(curl Q curl(!P 0 )) = (Q K1 )

for all Q 2 H 1 (G1): (4.3.15)

54

Thus, (4.3.3) with ( P ) replaced by (!0 !P 0 ), G replaced by G1 , implies ( for R

1 = G1 !P 0)

k!0 k2 G + k!P 0 ; 1 k1 G + tk!P 0 ; 1k2 G + t2k!P 0 ; 1 k3 G  C ; k!F 0 ; J(! 0 ) ; P 0 curl !k0 G + k!K 0 ; curl !  0 + ;1 t2  !P 0 + 2;1t2 grad !  grad P 0 k1 G  C ; kF k1 G + kK k2 G
: 1

1

1

1

1


1

Since function ! = 1 on G0 , inequality (4.3.4) is proved for s = 1. Now we are going to prove (4.3.4) for s = 2. The notation 00 now means either Dxx, or Dxy , or Dyy . Applying dierentiation rules, we can obtain (for the same ! as in before)

; div C E (!00) ; curl(!P 00 ) = !F 00 ; J(! 00 ) ; P 00 curl ! =: F2

(4.3.16)

; rot(!00 ) + ;1 t2 (!P 00) = !K 00 ; curl !  00

+ ;1 t2  !P 00 + 2;1 t2 grad !  grad P 00

=: K2

(4.3.17)

R

with G1 K2 = 0. Then, inequality (4.3.3), with ( P ) replaced by (!00 !P 00) and R G replaced by G1 , implies (for 2 = G1 !P 00 )

k!00 k2 G + k!P 00 ; 2k1 G + tk!P 00 ; 2k2 G + t2k!P 00 ; 2 k3 G  C ; k!F 00 ; J(! 00 ) ; P 00 curl !k0 G + k!K 00 ; curl !  00 + ;1 t2  !P 00 + 2;1 t2 grad !  grad P 00 k1 G  ; kF k2 G + kK k3 G + kk3 G + kP k2 G + tkP k3 G + t2kP k4 G
 C ; kF k2 G + kK k3 G
1

1

1

1

1


1

1

1

1

1

1

1

55

where in the last step we use (4.3.4) with s = 1 and G0 replaced by G1 . Since ! = 1 on G0, so inequality (4.3.4) is proved for s = 2. Same arguments, together with an induction on s could be used to prove (4.3.4) for s  3.

4.4. The Arnold-Falk Element Let Th denote a family of quasi-uniform triangulations of  and Pk (T ) the set of polynomials of degree not greater than k  0 restricted to T , an arbitrary element of Th . Consider the following nite element spaces

Qh = fq 2 L2 : qjT 2 P0 (T ) for all T 2 Th g

Ph = fp 2 H 1 : pjT 2 P1(T ) for all T 2 Th g P^h = Ph \ L^ 2

Wh = fw 2 L2 : wjT 2 P1(T ) for all T 2 Th and w is continuous at midpoints of element edgesg

W h = fw 2 L2 : wjT 2 P1(T ) for all T 2 Th and w is continuous at midpoints of element edges and vanishes at midpoints of boundary edgesg

V h = f
2 H 1 :
jT 2 P1(T ) B3(T )]2

for all T 2 Th g:

In the above, B3 (T ) is the cubic bubble function on T . For 0  , let V h(0 ) = f

j j 2 V h g V h (0) = f 2 V h j supp   0g Ph (0 ) = fqj j q 2 Ph g P h (0 ) = fq 2 Ph j supp q   0g

h (0 ) = fp 2 Wh j supp p   0g: Wh (0 ) = fpj j p 2 Wh g W Since Th is quasi-uniform, so the approximation property, superapproximation prop0

0

0

erty, and the inverse inequality property introduced in section 3.3 hold for the above nite element spaces. We will not repeat them here.

56

Let Ph0 be the local L2-projection operator onto Qh . Then the nite element in the primitive variables of Arnold-Falk (for the soft simply supported plate) reads as follows: Find (wh h ) 2 W h V h , such that ;

C E ( h ) E (
) + t;2 Ph0 h ; gradh wh
; gradh = g


;




;




(4.4.1)

for all (
) 2 W h V h . Then, under the discrete Helmhotz theorem of Arnold and Falk 3] Qh ]2 = gradh W h curl Ph (4.4.2) the discrete shear vector can be expressed as

 = t;2 (gradh wh ; Ph0 h ) = gradh rh + curl ph

(rh ph ) 2 W h P^h : (4.4.3)

Thus, equation (4.4.1) can be written equivalently in the form: Find (rh h ph wh) 2 W h V h P^h W h such that (gradh rh gradh ) = (g ) for all 2 W h

(C E ( h ) E (
)) ; (curl ph
) = (gradh rh
) for all
2 V h ;( h curl q) ; ;1 t2(curl ph curl q) = 0 for all q 2 P^h

(4.4.4) (4.4.5) (4.4.6)

(gradh wh gradh s) = ( h + ;1t2 gradh rh gradh s ) for all s 2 W h : (4.4.7)

The function rh is uniquely determined by (4.4.4). Since the MINI element is stable 2], i.e., there is a constant C , such that sup (curl p
)  sup (curl p
)  C kpk0
2Vh k
k1
2V h k
k1
6=0


6=0

57

for all p 2 P^h, so ( h ph ) is uniquely dened by equations (4.4.5) and (4.4.6), and thereafter, wh by equation (4.4.7). It is important to note that system (4.4.4)-(4.4.7) is for the purpose of convergence analysis only. Equation (4.4.1) is the one used for the actual computation. Our interior analysis of the Arnold-Falk element will also be based on the decoupled system of Poisson's equations and the Stokes-like equations, not the original Mindlin-Reissner plate system (i.e., (4.2.1)). Therefore, the interior estimate of the Arnold-Falk element consists of obtaining the interior estimate for the nonconforming element for the Poisson equation and that for the MINI element for the perturbed system (2.3.4) and (2.3.4). Since the rst part is done in Chapter 2, we need only concentrate on the Stokes-like system here. Before we turn to the next section, we introduce a result on the convergence of the MINI element for the perturbed Stokes-like system.

2 H 1(Gh), p 2 H 1(Gh ), 2 V h(Gh ) and p 2 Ph (Gh) with

Lemma 4.4.1. Let Gh a union of triangles. Then for

and F 2 L2 (Gh ), there exist unique functions

R R p = Gh Gh p, such that

; ) E (
)
; ; curl(p ; p)

= (F ;; ; curl q
; ;1 t2; curl(p ; p) curl q
= 0 ;

CE(


) for all
2 V h (Gh ) for all q 2 Ph (Gh ):

Moreover,

k ; k1 G + kp ; pk0 G + tk curl(p ; p)k0 G  C ; q2Pinf(G )(kp ; qk0 G + tk curl(p ; q)k0 G )
+ inf k ;
k1 G + kF k0 G :
2V (G ) h

h

h

h

h

h

h

h

h

h

h

(4.4.8)

58

Proof. The unique existence of solution ( p) follows from the stability property of the MINI element and Brezzi's Theorem 13] . The estimate (4.4.8) can be obtained by following the proof in 3, Theorem 5.5].

4.5 Interior Duality Estimates Let (w ) 2 H 1 H 1 be some solution to the Reissner-Mindlin plate equations and (r p) 2 H 1 H 1 be determined by the Helmholtz decomposition (4.2.2). Regardless of the boundary conditions used to specify the particular solution, (r p w) satises (grad r grad ) = (g ) for all 2 H 1

; ; curl p

= ; grad r

for all
2 H 1 ;( curl q) ; ;1t2(curl p curl q) = 0 for all q 2 H 1 (grad w grad s) = ( + ;1t2 grad r grad s) for all s 2 H 1:

;

C E ( ) E (
)




Similarly, regardless of the particular boundary conditions, the nite element solutions (rh h ph wh) 2 Wh V h Ph Wh satises

h (grad rh grad ) = (g ) for all 2 W

; ; curl ph

= ; gradh rh

for all
2 V h ;( h curl q) ; ;1t2(curl ph curl q) = 0 for all q 2 P h

h: (grad wh gradh s) = ( h + ;1t2 grad rh gradh s) for all s 2 W ;

C E ( h ) E (
)




59

Then, together with integration by parts, we obtain (gradh (r ; rh ) gradh ) = (C E (

;

E

X Z

T 2Th @T

@u v for all 2 W

h @n

; (curl(p ; ph )
) = (gradh (r ; rh )
) for all
2 V h ; ( ; h curl q) ; ;1 t2(curl(p ; ph ) curl q) = 0 for all q 2 P h

(4.5.1)

h ) (
))

(gradh (w ; wh ) gradh s) = (

;

X Z

T 2Th

@T

(

;

h + ;1 t2 gradh (r

@r )s  nT + ;1 t2 @n

(4.5.2) (4.5.3)

; rh ) gradh s)

h : for all s 2 W

(4.5.4)

As usual, the interior error analysis starts from these interior variational discretization equations. They are independent of the boundary conditions. The interior estimate for r ; rh is done in Chapter 2 (Theorem 2.5.2). However, we cannot use Theorem 2.5.2 directly to obtain the interior estimate for w ; wh. This is caused by the dierence between (2.4.1) and (4.5.4). But we can still use the same idea (as in the proof of Theorem 2.5.2) to get the interior estimate for (w ; wh ). This will be done in Theorem 4.6.3. So it is only necessary to obtain interior estimates for ( ; h p ; ph ) which satisfy (4.5.2) and (4.5.3). These will be the focus of this and next sections. What we will do is to use the same two-step approach (the interior duality estimate and the interior error estimate, as for the Stokes equations in Chapter 3) to obtain the interior estimate for the MINI element. Consider functions 2 H 1 and p 2 H 1 that satisfy the variational equations (C E ( ) E (
)) ; (curl p
) = 0 for all
2 V h

;( curl q) ; ;1t2(curl q curl p) = 0

for all q 2 P h :

(4.5.5) (4.5.6)

60

We have the following result.

Theorem 4.5.1. Assume 2 H 1 and p 2 H 1 satisfy (4.5.5) and (4.5.6). Let G0 b G b  be two concentric disks. Then for any integer   0, the following holds

k k0 G +kpk;1 G  C (hk k1 G +hkpk0 G +htk curl pk0 G +k k; G +kpk;;1 G): 0

0

(4.5.7)

Proof. Find a disk G1 such that G0 b G1 b G and construct a function ! 2 C01(G1 ) with ! = 1 on G0 . Then, for any non-negative integer s,

k k;s G  k! k;s G = 0

sup (k!F k F ) : sG  s (G) F 2H

(4.5.8)

F 6=0

To estimate the right hand side of (4.5.8), we dene ( P ) through (4.3.1) and (4.3.2) with K = 0. Then take  = ! in (4.3.1) to obtain

; ; ! curl P
;
;
; = C E ( ) E (!) ; curl(!P ) ; R(!  ) + curl ! P ;
;
= C E ( ) E (!)I ; curl(!P ) ;
;
+ C E ( ) E ! ; (!)I ] ; R(!  ) + curl ! P

;

! F = C E (! ) E ()


;

=: A1 + B1:







(4.5.9)

Here the superscript I is the approximation operator. Chosing
to be (! )I in

61

(4.5.5) we get ;

A1 = curl p (!)I




; ; curl(!P )


; ; curl(!P )
+ ; curl p (!)I ; !
;
;
= curl(!p)  ; curl(!P ) ; ; curl ! p
; ; curl p (!)I ; !
;

= curl p !




=: A2 + B2:

(4.5.10)

Taking Q = !p in (4.3.2) (with K = 0), we obtain

A2 = ;;1 t2 curl(!p) curl P

; ; curl(!P )
;
;
= ;;1 t2 curl p curl(!P ) ; curl(!P ) + ;1t2R0 (! P p) ;
;
= ; ;1 t2 curl p curl(!P )I + curl(!P ) ;


+ ;1t2 curl p curl(!P )I ; !P ] + ;1 t2R0 (! P p) =: A3 + B3: ;




(4.5.11)

Substituting q = (!P )I in (4.5.6), we have

A3 =

;

curl(!P )I




; ; curl(!P )
= ; curl(!P )I ; !P ]
:

(4.5.12)

Combining (4.5.9) through (4.5.12), we get ;




! F = B1 + B2 + B3 + A3 :

(4.5.13)

Then applying the approximation property, (4.2.7), (4.2.8), integration by parts, and the Schwarz inequality, we obtain

j B1 j C ; hk k1 G kk2 G + k k;s;1 G (kks+2 G + kP ks+1 G )
j B2 j C ; hkpk0 G kk2 G + kpk;s;2 G kks+2 G
j B3 j C ; ht2k curl pk0 G kP k2 G + t2kpk;s;2 G kP ks+3 G
j A3 j Chk k1 G kP k1 G : 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

(4.5.14)

62

First combining (4.5.14), (4.5.13), and (4.5.8), then applying (4.3.3) and (4.3.4), we obtain

k k;s G  C ; hk k1 G + hkpk0 G + k k;s;1 G + kpk;s;2 G + htk curl pk0 G
: 0

(4.5.15) R To estimate kpk;s;1 G0 , rst nd a function 2 C01(G1 ) with G = 1. Then,

kpk;s;1 G  k!pk;s;1 G = 0

Note that

(!p g) = (!p g ;

and

j (!p

Z

G

Z

G

(!p g) : g2H s+1 (G) kg ks+1 G sup

(4.5.16)

g6=0

g) + (!p

Z

G

g)

g) j C kpk;s;1 G  kgk0 G:

(4.5.17)

(4.5.18)

In order to estimate the rst term on the right hand side of (4.5.17), we dene ( P ) R through (4.3.1) and (4.3.2) with F = 0, K = g ; G g. Taking Q = !p in (4.3.2), we have ;

!p g ;

Z

g




G


= ; curl(!p) 

; ;1t2; curl(!p) curl P
;
;
;
= ; curl p ! ; ;1t2 curl p curl(!P ) ; curl ! p + ;1 t2R0 (! P p) ;
;
= ; curl p (!)I + ;1t2 curl p curl(!P )I ;
;
+ curl p (!)I ; ! + ;1 t2 curl p curl(!P )I ; !P ] ;; curl ! p
+ ;1t2R0 (! P p) ;

=: C1 + D1 :

(4.5.19)

63

Applying (4.5.5) and (4.5.6) with
= (!)I and q = (!P )I , respectively, we get

C1 = ; C E ( ) E (!)I + ;




;

curl(!P )I
;
;
;
= ; C E ( ) E (!) + curl(!P )I + C E ( ) E w ; (!)I ] ;
;
= ; C E (! ) E () ; curl(!P )I ;


+ C E ( ) E ! ; (!)I ] ; R(!  ) =: C2 + D2:

(4.5.20)

Taking  = ! in (4.3.1) (with F = 0), we obtain

C2 = ; ! curl P + ;




So far, we have

;

curl(!P )I
= ; curl(!P )I ; !P ]
+ ; curl ! P
: (4.5.21)

Z

(!p g ; g) = D1 + D2 + C2:

Then applying (4.2.7), (4.2.8), integration by parts, the approximation property, and Schwarz inequality, we arrive at

j D1 j C ; hkpk0 G kk2 G + ht2k curl pk0 GkP k2 G
+ kpk;s;2 Gkks+2 G + t2 kpk;s;2 GkP ks+3 G j D2 j C ; hk k1 G kk2 G + k k;s;1 G kks+2 G
j C2 j C ; hk k1 G kP k1 G + k k;s;1 G kP ks+1 G
: 1

1

1

1

1

1

1

1

1

1

(4.5.22)

1

1

Combining (4.5.16) through (4.5.22), together with (4.3.3) and (4.3.4), we obtain

kpk;s;1 G  C ; hk k1 G + hkpk0 G + k k;s;1 G + kpk;s;2 G + htk curl pk0 G
: 0

(4.5.23) Finally, (4.5.7) can be obtained by the standard iteration method (cf. section 2.4, section 3.4).

64

4.6 Interior Error Estimates In this section we rst obtain the interior estimate of the MINI element for the Stokes-like equations with perturbation, then we use it to derive the interior estimate for the Arnold-Falk element (Theorem 4.6.3). To be specic, Lemma 4.6.1 gives a bound on functions satisfying a homogeneous discrete Stokes-like equations. It is then used with Theorem 4.5.1 to get the interior estimate for the MINI element for the Stokes-like system (Theorem 4.6.2). By combining this result with the interior estimate of the nonconforming element (Theorem 2.5.2), we obtain the interior estimate of the Arnold-Falk element (Theorem 4.6.3). This is the main result of this chapter.

Lemma 4.6.1. Suppose (

h ph )

2 V h Ph is such that

(C E ( h ) E (
)) ; (curl ph
) = 0 for all
2 V h

;( h curl q) ; ;1 t2(curl ph curl q) = 0

for all q 2 P h :

(4.6.1) (4.6.2)

Then, for any two concentric disks G0 b G b , h small enough,  and  any nonnegative integers, we have

k hk1 G + kphk0 G + tk curl ph k0 G  C ; t (k hk1 G + tkphk1 G) + k h k; G + kph k;;1 G
0

0

0

(4.6.3)

where C = C (  G0 G): Proof. Let G0 b G0 b Gh b G1 b G with G0 a concentric disk and Gh a union of elements. Construct ! 2 C01(G0 ) with ! 1 on G0. Set fh = ! h , f ph = !ph . Then

65 fh

2 H 1(Gh), pfh 2 H 1(Gh ). By Lemma 4.4.1, fh 2 V h (Gh ) and fph 2 Ph (Gh )

can be uniquely determined by the equations

; ; curl(pfh ; fph)

= 0

for all
2 V h (Gh ) (4.6.4)

;; fh ; fh curl q
; ;1t2; curl(pfh ; fph ) curl q
= 0

for all q 2 Ph (Gh ) (4.6.5)

;

C E ( fh ; fh ) E (
)

R




R

with Gh f ph = Gh f ph . Moreover, we have

k fh ; fh k1 G + kpfh ; fph k0 G + tk curl(pfh ; fph)k0 G  C ;
2Vinf(G ) k fh ;
k1 G + q2Pinf(G )(kpfh ; qk0 G + tk curl(pfh ; q)k0 G )
 Ch; k hk1 G + kphk0 G + tkphk1 G
(4.6.6) h

h

h

h

h

h

h

h

h

h

h

h

h

where we have used the superapproximation property (cf. section 3.3) in the last step. By the triangle inequality

k hk1 G + kph k0 G + tk curl ph k0 G  k fhk1 G + kpfhk0 G + tk curl pfh k0 G  k fh ; fh k1 G + kpfh ; fph k0 G + tk curl(pfh ; fph )k0 G + k fh k1 G + k f ph k0 G + tk curl f ph k0 G  Ch(k hk1 G + kph k0 G + tkph k1 G ) + k fh k1 G + k f ph k0 G + tk curl f ph k0 G : 0

0

h

0

h

h

h

h

h

h

h

h

h

h

h

h

h

h

(4.6.7)

We shall consider k fh k1 Gh rst. In (4.6.4), we take
= fh to obtain ;

C E ( fh ) E ( fh ) = C E ( fh ) E ( fh )


;




; ; curl(pfh ; fph ) fh
:

(4.6.8)

66

We have ;

C E ( fh ) E ( fh ) = C E (! h ) E ( fh )


;

= C E ( h ) E (! fh )




; R(! fh ) = ; C E ( h ) E (! fh )I ]
n; o
I f f f + C E ( h ) E ! h ; (! h ) ] ; R(! h ) ;
=: C E ( h ) E (! fh )I ] + F1: ;




(4.6.9)

Taking
= (! fh )I in (4.6.1), we get C E ( h ) E (! fh )I ] =

;




;

curl ph (! fh )I
;
;
= curl ph ! fh + curl ph (! fh )I ; ! fh ;
n;
;
o I f f f f = curl(!ph ) h ; curl ! ph h ; curl ph (! h ) ; ! h ;
=: curl f ph fh + F2: (4.6.10)

Combining (4.6.8){(4.6.10) and substituting q = f ph in (4.6.5), we obtain ;

C E ( fh ) E ( fh ) =


;


curl f ph fh + F1 + F2 ;
;
= fh curl f ph + ;1 t2 curl(pfh ; f ph ) curl f ph + F1 + F2 ;
;
= ! h curl f ph + ;1 t2 curl(pfh ; f ph ) curl f ph + F1 + F2 ;
;
;
= h curl(! f ph ) ; curl ! f ph h + ;1 t2 curl(pfh ; f ph ) curl f ph

+ F1 + F2 =

;

+

h

;

curl(! f ph )I + ;1 t2 curl(pfh ; f ph ) curl f ph
;
ph ; (! f ph )I ] ; curl ! f ph h + F1 + F2 h curl! f


=: E1 + F3 + F1 + F2:

;




(4.6.11)

67

Setting q = (! f ph )I in (4.6.2), we get

E1 = ;;1t2 curl ph curl(! f ph )I + ;1t2 curl(pfh ; f ph ) curl f ph ;




;




= ;;1t2 curl ph curl(! f ph ) + ;1t2 curl ph curl! f ph ; (! f ph )I ] ;




;

+ ;1 t2 curl(pfh ; f ph ) curl f ph ;







= ;;1t2(curl(!ph ) curl f ph ) + ;1 t2 curl(pfh ; f ph ) curl f ph ;




+ ;1 t2 curl ph curl! f ph ; (! f ph )I ] + ;1t2 R0 (! ph f ph )

;




=: ;;1t2(curl f ph curl f ph ) + F4 :

(4.6.12)

So far, we have

kC E ( fh )k20 G 1 2

h

+ ;1 t2k curl f ph k20 Gh = F1 + F2 + F3 + F4:

(4.6.13)

Using the superapproximation properties (cf. section 3.3), the Schwarz inequality, integration by parts, (4.2.7), and (4.2.8), we obtain

j F1 j C ; hk hk1 G k fh k1 G + k h k0 G k fh k1 G
j F2 j C ; kph k;1 G k fh k1 G + hkphk0 G k fh k1 G
j F3 j C ; hk hk1 G k fph k0 G + k hk0 G k fph k0 G
j F4 j Ct2; hk curl ph k0 G k fph k1 G + kphk1 G k fphk0 G h

h

h

h

h

h

h

h

h

h

h

h

h

h

h


h

:

Combining the above inequalities with (4.6.13), using the inverse inequalities for

f ph , h , and ph , we get

k fhk21 G + t2k curl fphk20 G  C ; hk hk1 G + k h k0 G + hkphk0 G + kphk;1 G
k fh k1 G ;
+ C hk h k1 G + k h k0 G + t2kph k1 G k f ph k0 G  C ; k h k0 G + kphk;1 G
k fh k1 G + C ; k h k0 G + t2kph k1 G
k fphk0 G : h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

h

(4.6.14)

68

To proceed, we need to estimate k f ph k0 Gh . By the triangle inequality,

ph k Gh f k fphk0 Gh  k fph ; meas( k 0 G h + Gh )

ph ; pfh k0 Gh k Gh f ph k0 Gh Gh f + meas(Gh ) meas(Gh ) : (4.6.15) It is easy to see that the second term on the right hand side of the above inequality is bounded by the right hand side of (4.6.6). For the last term we have R

k

Z

Gh

k

ph 0 Gh f

=k

Z

Gh

R

R

!ph k0 Gh  C kphk;1 Gh :

From that fact that the triangulation Th is quasi-uniform, we have the following stability condition for the MINI element on set Gh , ;

R

curl f ph
ph Gh f k fph ; meas( k 0 Gh  C sup Gh ) k
k1 Gh
2V h (Gh )




Gh :

(4.6.16)


6=0

Applying (4.6.4), we obtain

; ; C E ( fh ; fh ) E (
)
;
;
= curl(!ph )
; C E ( fh ; e) E (
) ;
;
;
= curl ph w
; C E ( fh ; e) E (
) + curl ! ph
n;
;
o = curl ph (!
)I ; C E ( fh ; e) E (
) ;
;
+ curl ! ph
+ curl ph !
; (!
)I =: G1 + H1 : ;

(curl f ph
) = curl f ph





(4.6.17)

69

Setting
= (!
)I in (4.6.1), we get

G1 = C E ( h ) E (!
)I

; ; C E ( fh ; e) E (
)
;
;
= C E ( h ) E (!
) + C E ( h ) E (!
)I ; !
] ; ; C E ( fh ; e) E (
)
n;
;
o f e = C E (! h ) E (
) ; C E ( h ; ) E (
) ;
+ R(!
h ) + C E ( h ) E (!
)I ; !
] ;
=: C E ( fh ) E (
) + H2: ;




(4.6.18)

Applying the superapproximation property, the Schwarz inequlity, (4.2.7), (4.2.8), and integration by parts, we have

j H1 j C ; kphk;1 G k
k1 G + hkph k0 G k
k1 G
j H2 j C ; k h k0 G k
k1:G + hk hk1 G k
k1 G
j (E ( fh ) E (
)) j k fh k1 G k
k1 G : h

h

h

h

h

h

h

h

h

h

Combining (4.6.6), (4.6.15){(4.6.18), the above inequalities, and using the inverse inequalities, we arrive at

k fphk0 G  C ; hk hk1 G + k h k0 G + htkphk1 G + hkph k0 G + kph k;1 G ) + k fh k1 G  C ; k h k0 G + htkphk1 G + kph k;1 G
+ k fh k1 G : h

h

h

h

h

h

h

h

h

h

h

(4.6.19) Substituting (4.6.19) into (4.6.14) and using the arithmetic-geometric mean inequality, we have

k fh k1 G

h

+ tk curl f ph k0 Gh  C

;

k hk0 G

h

+ (t2 + ht)kph k1 Gh + kph k;1 Gh : (4.6.20)


70

Substituting (4.6.20) back into (4.6.19), we get

k fph k0 G  C ; k hk0 G + kph k;1 G h

h

h

+ (ht + t2 )kph k1 Gh :


(4.6.21)

Hence, combining (4.6.20) and (4.6.21) with (4.6.7), we obtain

k hk1 G + kph k0 G + tk curl phk0 G (4.6.22)  C ; k hk0 G + kph k;1 G + (ht + t2)kph k1 G
: Applying Theorem 4.5.1 with G0 replaced by G1 to bound k h k0 G and kphk;1 G , 0

0

0

h

h

h

1

1

we get

k hk1 G + kphk0 G + tk curl ph k0 G  C ; hk hk1 G + hkphk0 G + t(t + h)kph k1 G + k h k; G + kphk;;1 G
: Iterating the above inequality  ; 1 times as in section 2.5, or section 3.5, but separating the case h  t with that h  t, we prove (4.6.3). Theorem 4.6.2. Let 0 b 1 b . Suppose that ( p) 2 H 1 H 1 satises j 2 H 2(1 ) and pj 2 H 2(1). Suppose that ( h ph ) 2 V h Ph are such that 0

0

1

0

1

(C E (

;

E

; (curl(p ; ph )
) = (F
) for all
2 V h (4.6.23) ;( ; h curl q) ; ;1 t2(curl(p ; ph) curl q) = 0 for all q 2 P h : h ) (
))

(4.6.24)

for some function F in L2. Let  and  be two arbitrary nonnegative integers. Then, there is a positive number h1 such that for h 2 (0 h1 ],

k ; hk1  + kp ; phk0  + tk curl(p ; ph)k0   C ; kF k0  + h(k k2  + kpk1  + tkpk2  ) + t k ; h k1 
+ t+1kp ; ph k1  + k ; h k;  + kp ; ph k;;1  0

0

1

1

1

0

1

1

1

1

1

(4.6.25)

71

for a constant C depending only on 1, 0, , and  . Proof. Let G0 b G00 b G0 b Gh b G1 b G be concentric disks and nd a ! 2

1 (Gh ), pe 2 H 1(Gh ). C01(G0 ) with ! 1 on G00 . Set e = ! , pe = !p. Then e 2 H By Lemma 4.4.1, e 2 V h(Gh ), pe 2 Ph(Gh ) can be dened uniquely by the following equations, ;

C E ( e ; e) E (
)




; ; curl(pe ; pe)

= (F


) for all
2 V h (Gh ) (4.6.26)

;; e ; e curl q
; ;1t2; curl(pe ; pe) curl q
= 0 R

for all q 2 Ph (Gh ) (4.6.27)

R

with Gh pe = Gh pe. Moreover, we have

k e ; ek1 G + kpe ; pek0 G + tk curl(pe ; pe)k0 G  C ; q2Pinf(G )(kpe ; q k 0 G + tk curl(pe ; q)k0 G ) + kF k0 G
+ inf k e ;
k1 G
2V (G )  C ; k k1 G + kpk0 G + tkpk1 G + kF k0 G
: h

h

h

h

h

h

h

h


h

h

h

(4.6.28)

Let us now estimate k ; h k1 G0 , kp ; phk0 G0 , and tk curl(p ; ph)k0 G0 . By the triangle inequality, we have

k ; hk1 G + kp ; ph k0 G + tk curl(p ; ph )k0 G  k ; ek1 G + kp ; pek0 G + k e ; h k1 G + k pe ; phk0 G + tk curl(p ; pe)k0 G + tk curl( pe ; ph )k0 G : 0

0

0

0

0

0

0

0

0

Since (4.6.26), (4.6.27) and (4.6.23), (4.6.24) hold for any


2

(4.6.29)

V h (G00),

72

q 2 P h (G00 ), respectively. Subtracting the corresponding two equations, we obtain

(E ( h ; e) E (
)) ; (curl(ph ; pe)
) = 0 for all
2 V h (G00 ) (4.6.30) ;( h ; e curl q) ; ;1 t2(curl(ph ; pe) curl q) = 0 for all q 2 P h (G00): (4.6.31)

Then we apply Lemma 4.6.1 to h ; e and ph ; pe with G replaced by G00 to obtain

k h ; ek1 G + kph ; pek0 G + tk curl(ph ; pe)k0 G  C ; t k h ; ek1 G0 + t+1kph ; pek1 G0 + k h ; ek; G0
+ kph ; pek;;1 G0  C ; t+1kp ; pek1 G0 + t+1kph ; pk1 G0 + k ; ek; G0 + k h ; k; G0
+ t k ; h k1 G0 + t k h ; h k1 G0 + kp ; pek;;1 G0 + kp ; ph k;;1 G0  C ; k e ; ek1 G + kpe ; pek0 G + tk curl(pe ; pek0 G
+ t k ; h k1 G + t+1kp ; ph k1 G + k ; h k; G + kp ; ph k;;1 G : 0

0

0

0

0

0

0

0

0

0

0

0

h

0

h

0

0

h

(4.6.32)

Combining (4.6.28), (4.6.29), and (4.6.32), we obtain k ; hk1 G0 + kp ; phk0 G0 + tk curl(p ; ph)k0 G0

 C ; kF k0 G + k k1 G + kpk0 G + tk curl pk0 G
+ t k ; h k1 G + t+1kp ; ph k1 G + k ; h k; G + kp ; ph k;;1 G : Since ( ; q) ; ( h ; q)] and (p ; q) ; (ph ; q)] also satisfy equations (4.6.23) and (4.6.24) for any
2 V h and q 2 P h , we have k ; hk1 G + kp ; phk0 G + tk curl(p ; ph)k0 G  C ;
inf k ;
k1 G + qinf (kp ; qk0 G + tk curl(p ; q)k0 G) + kF k0 G 2V 2P
+ k ; h k; G + kp ; ph k;;1 G + t k ; h k1 G + t+1kp ; ph k1 G : 0

h

0

0

h

73

Then, rst using the approximation properties of the nite element spaces and then a covering argument (cf. section 2.5 and section 3.5), we obtain the desired result. We now state the main result of this chapter.

Theorem 4.6.3. Let 0 b 1 b  and suppose that (r p w) 2 H 1 H 1 H 1

H 1 (the exact solution) satises (r p w)j1 2 H 2 (1 ) H 2(1 ) H 2(1 ) H 2 (1). Suppose that (rh h ph wh) 2 Wh V h Ph Wh (the nite element solution) is given so that (4.5.1), (4.5.2), (4.5.3), and (4.5.4) hold. Let ,  be two nonnegative integers. Then there exists a positive number h1 and a constant C depending only on 0, 1, , and  , such that for all h 2 (0 h1 ]

kr ; rhkh1   C ; hkrk2  + kr ; rh k; 
(4.6.33) k ; hk1  + kp ; phk0  + tk curl(p ; ph)k0   C ; h(k k2  + kpk1  + tkpk2  + krk2  ) + kr ; rhk; 
+ k ; h k;  + kp ; phk;;1  + t k ; h k1  + t+1kp ; ph k1  1

0

0

1

0

1

0

1

1

1

1

1

1

1

1

kw ; whkh1   ; Ch(k k2  + kpk1  + tkpk2  + krk2  + kwk2  ) + kw ; whk;  + k ; h k;  + kr ; rh k; 
+ kp ; ph k;;1  + t k ; h k1  + t+1kp ; ph k1  : 1

0

1

1

1

1

1

1

(4.6.34)

1

1

1

1

(4.6.35)

Proof. Find a subdomain 0 such that 0 b 0 b 1. Apllying Theorem 2.5.2 with 0 replaced by 0 yields

kr ; rhkh1 0  C ; hkrk2  + kr ; rh k;  1


1

(4.6.36)

which also implies (4.6.33). From (4.5.2), (4.5.3), (4.6.36), and Theorem 4.6.2 with

74

1 replaced by 0 , we obtain

k ; h k1  + kp ; phk0  + tk curl(p ; ph )k0   C ; h(k k2 0 + kpk1 0 + tkpk2 0 ) + k gradh (r ; rh )k0 0
+ k ; h k; 0 + kp ; ph k;;1 0 + t+1kp ; ph k1 0 + t k ; h k1 0  C ; h(k k2  + kpk1  + tkpk2  + krk2  ) + k ; hk; 
+ kp ; ph k;;1  + t k ; h k1  + t+1kp ; ph k1  + kr ; rh k;  : 0

0

1

0

1

1

1

1

1

1

1

1

This completes (4.6.34). We now consider the interior estimate for the transverse displacement. Because of the dierence between (2.4.1) (required by Theorem 2.5.2)) and (4.5.4) (satised by (w ; wh)), Theorem 2.5.2 cannot be used directly. But we can follow the same proof to get (4.6.35). Let G0 b G1 b G be concentric disks and Gh be a union of triangles which satises G1 b Gh b G. Use the notation we = !w and dene we 2 W h (Gh ) by (gradh we gradh s) = (gradh we gradh s) for all s 2 W h (Gh ): We have

kwe ; wekh1 G  C s2W inf(G ) kwe ; sk1 G  C hkwk2 G : h

h

h

h

h

By the triangle inequality,

kw ; whkh1 G  kwe ; wekh1 G + k we ; wh kh1 G  Chkwk2 G + k we ; whkh1 G : h

0

h

0

0

(4.6.37)

From (4.5.4) and the fact that ! = 1 on G1, (gradh ( we ; wh ) gradh s) = (gradh (w ; wh) gradh s)

h (G1 ) = L(s) for all s 2 W

(4.6.38)

75

where

;

L(s) := (

h + ;1 t2 grad

h (r

; rh ) gradh s) ;

X Z

T 2Th @T

@r )s  nT + ;1t2 @n

(

h . By Lemma 2.3.2, for all s 2 W

jL(s)G j  C; k ; hk1 G + kr ; rh kh1 G
+ h(k k1 G + krk2 G ) k gradh sk0 G h

h

h

h

h

h

h (Gh ) for all s 2 W

which implies

kLkG  C ; k ; hk1 G + kr ; rh kh1 G h

h

h

+ h(k

k1 G + krk2 G )
: h

h

Under (4.6.38) we apply Lemma 2.5.1 with G replaced by G1 to obtain

k we ; whkh1 G  C ; k we ; whk;t G + kLkG )  C (k we ; wek;t G + kw ; wh k;t G + kLkG )  C ; k ; h k1 G0 + kr ; rh kh1 G0 + kw ; whk; G
+ h(k k1 G + krk2 G ) : 1

0

h

1

1

h

(4.6.39)

Using the triangle inequality, (4.6.37), and (4.6.39) yields

kw ; whkh1 G  C ; k ; h k1 G0 + kr ; rh kh1 G0 + kw ; whk; G
+ h(k k1 G + kwk2 G + krk2 G ) : 0

Applying (4.6.33) and (4.6.34) with 0 and 1 replaced by G0 and G, respectively, we obtain a local version of (4.6.35). Then a standard covering argument leads to (4.6.35).

76

4.7 The Global and Interior Convergences of the Arnold-Falk Element As an application of the theory we developed in the last section, we consider the soft simply supported plate with a smooth forcing function g. Under this boundary condition (for a smooth ), the exact solution of the Reissner-Mindlin plate satises (cf. 5]) krk2 + kwk2 + k k3=2 + kpk1=2 + tkpk3=2  C

k k3=2+ + kpk1=2+ + tkpk3=2+  Ct; k k2 + kpk1 + tkpk2  Ct;1=2 for a constant C that is independent of t and h, and  2 (0 1=2].

(4.7.1)

Obviously, functions and p are not regular enough to ensure that the MINI element converges at the optimal rate, uniformly in the plate thickness t. Thus, we want to use Theorem 4.6.3 to obtain the interior convergence rates of the Arnold-Falk element. To do so, we must estimate kp ; phk1 , k ; h k1, kr ; rhk; 1 , k ; hk; 1 , kp ; ph k;;1 1 , and kw ; wh k; 1 for some suitable domain 1 and integer . The only way of doing these, as far as we know, is to use the inequality k  kt 1  k  kt  . Hence, the global convergence of the nonconforming element for the Poisson equation and that of the MINI element for the Stokes-like equations must be established. The rst one is well-known but the second one is dicult due to the special structure of the singularly perturbed Stokes-like equations and the type of the boundary condition imposed. Because of the diculty in dealing with the boundary approximation when the boundary layer exists, we will only work on a polygonal domain, i.e., we will assume that  is a convex polygon. Therefore, we need to know the regularity of the exact solution of the singularly perturbed generalized Stokes-like equations on a convex polygon (under the soft simply supported boundary condition). But so far we cannot prove

77

the regularity result (Theorem 4.7.2) we need. So we will assume that it is true. We feel that we have reason to believe it is correct (possibly with some restriction on the magnitude of the maximum angle of the polygon). As a partial justication, we will prove a similar result for a smooth  in Appendix B. This section is organized as follows. Theorem 4.7.1 presents a technical result on the approximation property of the continuous piecewise linear functions. Its proof can be found in Appendix A. Theorem 4.7.2 is the assumption we just mentioned. The global convergence of the Arnold-Falk element is given in Theorem 4.7.3. Finally by combing Theorem 4.7.2 and Theorem 4.6.2 we get the interior convergence rate of the Arnold-Falk element for the rotation (Theorem 4.7.4).

Theorem 4.7.1. Let  be a convex polygon and u 2 H 2 . Then there exists an operator h : H 1 ! Ph such that

kp ; h pk0  Ch1=2+kpk1=2+ kp ; h pk1  Ch1=2kpk3=2 kp ; h pk;1=2 @  Ch1=2+kpk1=2+

(4.7.2) (4.7.3) (4.7.4)

for any 0 <   1=2. Here C is independent of  and C depends on , but not h. Proof. See Theorem 5.3.1, Appendix A.

Theorem 4.7.2. Let  be a convex polygon and F 2 H 1 and K 2 H 2 \ L^ 2. Then there exists a unique solution ( P ) 2 H 2 H 1 \ L^ 2 to the equations (C E () E ()) ; ( curl P ) = (F ) for all  2 H 1

;( curl Q) ; ;1t2 (curl Q curl P ) = (K Q)

for all Q 2 H 1 :

(4.7.5) (4.7.6)

78

Moreover,

kk3=2+ + kP k1=2+ + tkP k3=2+  Ct;; kF k0 + t3=2kF k1
kk3=2+ + kP k1=2+ + tkP k3=2+  Ct;kK k2 for 0    1=2.

for K = 0 (4.7.7)

for F = 0

(4.7.8)

Proof. See Corollary 6.4, Appendix B.

Theorem 4.7.3. Let  be a convex polygon. Assume that (r p w) and (rh h ph wh ) solve (4.2.3){(4.2.6) and (4.4.4){(4.4.7), respectively, for some smooth g, some t 2 (0 1], and a quasi-uniform mesh Th . Then,

k ; h k1 + kp ; ph k0 + tk curl(p ; ph )k0  Ch1=2t; k ; h k0 + kw ; whkh1  Cht; tk ; h k1 + t2k curl(p ; ph )k0  Ch k ; h k;1 + kp ; ph k;2  Cht;

(4.7.9) (4.7.10) (4.7.11) (4.7.12)

for an arbitrarily small constant . Proof. Subtracting (4.2.4) by (4.4.5) and (4.2.5) by (4.4.6), respectively, we obtain ;

C E(

;

E

h ) (
)

;; curl q ;




; ;
curl(p ; ph )
= ; grad(r ; rh )

for all
2 V h h




(4.7.13)

; ;1 t2; curl(p ; ph ) curl q
= 0 for all q 2 Ph

(4.7.14) From (4.7.13) and (4.7.14) (see also the proof on page 1284 of 3, Theorem 5.5]), we have kC 21 E ( h ;
)k20 + ;1t2k curl(ph ; q)k20 = C E(

;
) E ( h ;
)
+ ;1t2; curl(p ; q) curl(ph ; q)
; ; curl(p ; q) h ;

+ ; ;
curl(ph ; q)
+ ; gradh (rh ; r) h ;

: ;

(4.7.15)

79

In the above we choose
to be the Fortin projection of , that is,

; curl q) = 0 k ; k1  C k ;
k1 (

for all q 2 Ph

for all
2 V h :

We see that the fourth term on the right hand side of (4.7.15) is gone. Taking q = h p to be the interpolant of p described in Theorem 4.7.1 and using the Schwartz inequality, integration by parts, and the arithmetic-geometric mean inequality, we obtain

kC E ( h ; )k20 + ;1t2k curl(ph ; h p)k20  C (k ; k21 + t2kp ; h pk21 + kp ; h pk0k rot( h ; )k0 + k h ; k1kp ; h pk;1=2 @ + hk h ; k1 ): 1 2

To estimate k h ; is equivalent to

(4.7.16)

k1, we note that the H 1 norm of the vector function h ; Z

Z

kE ( ; h )k0 + j ; h j + j ( ;

 ;y x)j:

h) (

(4.7.17)

The rst term in (4.7.17) is already covered by (4.7.16). To control the other two, we take q = y and q = x, respectively in (4.7.14) to get

j

Z

Z

; h )j  j curl(p ; ph )j Ct2

(

which implies Z

Z

j ( ; h)j  j ( ; )j + Ct2k curl(p ; ph )k0  C ; k ; k0 + t2k curl(p ; h p)k0 + t2k curl( h p ; ph )k0
:

(4.7.18)

80

To control the third term in (4.7.17), we take q = Lh q0 = Lh (x2 + y2)=2: the L2 projection of (x2 + y2 )=2 in Ph() in (4.7.14), to obtain ;

; h curl q0
= ; ; h curl(q0 ; Lh q0)
; ;1t2; curl(p ; ph ) curl Lhq0
:

Using the fact that and

kq0 ; Lh q0 k1 1  Ch kLh q0k1 1  C

we obtain

j

Z

(

;

 ;y x)j  C hk ; h k0

h) (

;

+ t2

Z

j curl(p ; ph)j
:

(4.7.19)

Therefore, Z

j ( ; h )  (;y x)j  C ; k ; k0 + hk ; h k0 + t2k curl(p ; hp)k0 + t2k curl( hp ; ph )k0
:

(4.7.20)

Combining (4.7.16){(4.7.20) and using the arithmetic-geometric mean inequality, we get

k h ; k1 + tk curl(ph ; h p)k0  C ; k ; k1 + tkp ; h pk1 + kp ; hpk0 + kp ; hpk;1=2 @ + h
: Applying the approximation property of and Theorem 4.7.1, we get

k h ; k1 + tk curl(ph ; h p)k0  Ch1=2; k k3=2 + tkpk3=2 + kpk1=2+ + h1=2


81

which implies

k ; h k1 + tk curl(p ; ph )k0  Ch1=2t;:

(4.7.21)

Moreover,

tk

; h k1 + t2k curl(p ; ph )k0  Ct; k ; k1 + tkp ; h pk1
+ kp ; h pk;1=2 @ + h  Cth; k k2 + tkpk2 + 1
 Ch:

To estimate kp;ph k0, we simply repeat the proof on page 1285 of 3, Theorem 5.5]. By the stability condition, there exits  > 0 independent of h such that for all q 2 W^ h there exists a nonzero
2 V h with

 kqk0k grad
k0  (curl q
): Applying this result with q replaced by (ph ; h p + and (4.4.6), we have

 kph ; h p +

Z



R

 h p), and again using (4.2.4)

h pk0k grad
k0  curl(ph ; h p)
;

= curl(p ; h p)
+ C E ( h ; ) E (
)




; ; gradh(rh ; r)

 C ; kp ; hpk0 + k grad( h ; )k0 + k gradh (rh ; r)k0
k grad
k0 ;




;




so

kph ; hp +

Z

h pk0  C kp ; h pk0 + k ;



; h k1 + k gradh(r ; rh )k0
:

By the triangle inequality, we get

kp ; ph k0  C ; kpk0 + kp ; h pk0 + k ; h k1 + k gradh(r ; rh )k0
:

82

Since if ( ; h p ; ph ) satises (4.7.13) and (4.7.14), so will ( ;
) ; ( h ;
) (p ; q) ; (ph ; q)], for any (
q) 2 V h Wh. Therefore, together with (4.7.21) and Theorem 4.7.1, we get

kp ; phk0  C ; kp ; h pk0 + k ; h k1 + k gradh(r ; rh )k0
 Ch1=2t;: This completes (4.7.9). To bound k ; h k0, we construct the following duality problem: Find ( P ) 2 H 1 H^ 1 such that

(C E (
) E ()) ; (
curl P ) = (

;(curl q ) ; ;1t2(curl q curl P ) = 0

;

h
) for all


2 H1

for all q 2 H 1:

(4.7.22) (4.7.23)

From Corollary 4.2.4 we know that ( P ) is uniquely dened. Moreover,

kk3=2+ + kP k1=2+ + tkP k3=2+  Ct;; k ; hk0 + t3=2k ; h k1
(4.7.24) Following the proof of Theorem 6.1 on page 1286 of 3], we can obtain, for (q
) = ( h P ), where is the Fortin projection and h the interpolant as described in Theorem 4.7.1,

k ; h k20 =; C E ( ; h) E ( ; )
; ; ; h curl(P ; h P )
; ; curl(p ; hp)  ; 
; ;1t2; curl(p ; ph) curl(P ; h P )
;
+ gradh (r ; rh )  ;
;
= C E ( ; h ) E ( ; ) ; rot( ; h ) P ; h P ;h( ; h )  s P ; h P i@ + ; p ; h p rot( ; )
;hp ; hp ( ; )  si@ ; ;1 t2; curl(p ; ph) curl(P ; h P )
;
+ gradh (r ; rh )  (4.7.25)

83

Using the Schwartz inequality and integration by parts in (4.7.25), we get

k ; h k20  C k ; h k1(k ; k1 + kP ; h P k0 + kP ; h P k;1=2 @ ) + k ; k1(kp ; h pk0 + kp ; h pk;1=2 @ ) + t2k curl(p ; ph )k0 k curl(P ; h P )k0 + k gradh (r ; rh )k0 k k0 :

(4.7.26)

If t  h, then by Theorem 4.7.1, (4.7.26), (4.7.24), and (4.7.9), we obtain

k ; hk20  Cht;; kk3=2 + kP k1=2+
+ Chkk3=2kpk1=2+ + ChkP k3=2 + Chkk1  Cht;2; k ; hk0 + t3=2k ; h k1
 Cht;2; k ; hk0 + h2
which implies

k ; h k0  Cht;2:

If t  h, using (4.7.24) with  = 1=2 in (4.7.26), we obtain

k ; hk20  Ch3=2t; (kk2 + kP k1) + Ch3=2t;kk2 + Ch3=2t; kP k2 + Chkk1  Ch3=2t; t;1=2(k ; h k0 + t3=2 k ; h k1) + Chk ; h k0: Since t  h, Hence

k ; h k20  Cht;(k ; h k0 + h2): k ; h k0  Cht;:

84

To analyze w ; wh , we follow that in 3,Theorem 5.5]. Dene !h 2 W h by ;

gradh wh gradh s
= ; + ;1t2 grad r gradh s
for all s 2 W h:

So

kw ; whkh1  Ch; k + t2rk1 + kwk2
 Ch where we use the fact that k k1 + krk1 + kwk2  C , with C independent of t. In addition, we have ;

gradh (wh ; wh) gradh s =


;

;

h + ;1 t2 gradh (r

; rh) gradh s


for all s 2 Wh . Obviously,

kwh ; whkh1  C ; k ; h k0 + t2k gradh (r ; rh )k0
 C ; h + k ; hk0
which implies

kw ; whkh1   kw ; wh kh1 + kwh ; whkh1  C (h + k ; h k0)  Cht;: This completes (4.7.10). We will use the duality argument again to estimate kp ; phk;2 . To do so, we introduce the following auxiliary problem. Find ( P ) 2 H 1 H^ 1 such that (C E (
) E ()) ; (
curl P ) = 0

for all
2 H 1 (4.7.29)

;(curl q ) ; ;1t2(curl q curl P ) = (K q) for all q 2 H 1 (4.7.30) for any K 2 L^ 2. By Corollary 4.2.4, this problem admits a unique solution, and that

kk3=2+ + kP k1=2+ + tkP k3=2+  Ct;kK k2:

(4.7.31)

85

By denition,

kp ; ph k;2 =

sup (p ; ph K ) : K 2H 2 kK k2

(4.7.32)

K 6=0

In (4.7.30), we take q to be p ; ph and apply (4.2.4), (4.2.5), (4.4.5), (4.4.6), and (4.7.29) to obtain (p ; ph K ) = ;  ;  curl(p ; ph )

; ;1t2; curl(p ; ph) curl(P ; hP )
; ; C E ( ; h ) E ( ; )
+ ; ; h curl( h P ; P )
;
+ gradh (r ; rh )  (4.7.33) ;




Using the Schwartz inequality, Theorem 4.7.1, and (4.7.31) in (4.7.33), we get

j(p ; ph K )j  Cht;; kk3=2 + tkP k3=2 + kP k1=2+
:

(4.7.34)

Combining (4.7.31){(4.7.34), we arrive at

kp ; phk;2  Cht;2: Similarly, we can prove

k ; h k;1  Cht;2:

Since  is an arbitrary number, then (4.7.12) is proved. Equipped with the above result, we are able to prove the following interior estimate for the Arnold-Falk element for the rotation .

Theorem 4.7.4. Let  be a convex polygon and 0, b  an interior domain. Let g

be a smooth function. Assume that Th is quasi-uniform. Suppose that (w ) solves (4.2.1) and (wh h ) solves (4.4.1). Then there exists a number h1  0, such that for all h 2 (0 h1 ], k ; h k1 0  C ht; (4.7.35)

86

where C is independent of t and h. Proof. First choose 1 such that 0 b 1 b . Then note that k  ks 1  k  ks . Combining Theorem 4.7.3 with Theorem 4.6.3 with  = 2,  = 2, (4.7.35) can be obtained.

Because the Brezzi-Fortin element (cf. 15]) is also based on the variational formulation (4.2.3)-(4.2.6), we have the following result.

Corollary 4.7.5. Assume that the Brezzi-Fortin method 15] is used to solve (4.2.3){ (4.2.6). Then, under the same conditions of Theorem 4.7.4, Theorem 4.7.3 and Theorem 4.7.4 hold.

4.8 Numerical Results In this section we give the results of computations of the solutions to the ArnoldFalk element for the Reissner-Mindlin plate model. Through a model problem, we show that the Arnold-Falk approximation for rotation does not achieve the global rst order convergence rate in the energy norm for the soft simply supported plate, but it does have rst order convergence rate for the transverse displacement w. We will also show that the Arnold-Falk method obtains the rst order convergence rate for the rotation in the region away from the boundary layer. Thereafter, numerical computations conform to the theoretical predictions. We will take the domain  to be the unit square. Since we know the exact solution of the semi-innite (y > 0) Reissner-Mindlin plate when the load function g(x y) = cos(x) and the plate is soft simply supported on the boundary y = 0, we can simply restrict this solution to . By doing so, we obtain the exact solution of the unit square plate with the hard clamped boundary condition on the left, upper, and right edges, and soft simply supported boundary condition on the lower edge

87

(0 < x < 1, y = 0) (cf. 8]). And the lower edge (0 < x < 1, y = 0) is where the boundary layer occurs. We take E = 1,  = 3=10, and  = 5=6. Moreover, the mesh is taken to be uniform. The interior domain is taken to be the upper half of the unit square (since the boundary layer only exists near the lower edge 0 < x < 1, y = 0). All computations were performed on a Sun SPARCStation 2 using the Modulef (INRIA) package. A distinguished feature of this test problem is that the exact solution has the following property: 1 2 H 3=2 () and 2 2 H 5=2 (), i.e., 1 has a stronger boundary layer than 2 does (cf. 5]). The numerical results unmistakenly express this dierence. In each graph (of Figure 4.1{Figure 4.6) the H 1 norms of the errors on the global domain and the interior domain, are plotted as functions of the mesh size h. The values of h are 1=2, 1=4, 1=6, 1=8, 1=10, 1=12, 1=16, 1=20, and 1=24. Both axes have been transformed logarithmically so that the slope of the error curves gives the apparent rate of convergence as h tends to zero. Absolute errors are shown. Figures 4.1{4.2 show the approximation errors in H 1 norm of 1 and 2 for t = 1 and the rst order optimal convergence rate is as expected. And there is no dierence between the rate on the whole domain and that on the upper half unit square. Figures 4.3{4.4 show the errors in H 1 norm of 1 and 2 for t = 0:0001. It is clear that when t is small, the boundary layer eect of 1 comes into play and as a result, we only see a 1=2 order convergence rate for k1 ; h1 k1 on the whole domain. However, away from the boundary layer, the optimal rst order convergence rate is recovered. Figures 4.5{4.6 show the errors in H 1 norm of the transverse displacement w for t = 1 and 0:0001. In all cases, the rst order convergence rate is observed, because

88

there is no boundary layer in the transverse displacement. Figures 4.7{4.12 show the errors in L2 norm for variables 1 , 2, and w, with the thickness of the plate t = 1 and t = 0:0001, respectively. We note that in the interior domain, the optimal convergence rate (second order) is observed, but this cannot be proved by the current method. (Though we did not explicitly state a theorem about the interior estimate in the L2 norm in section 4.6, it is not dicult to do so in the light of Chapter 2 and Chapter 3.) The global convergence rates in the L2 norm are also higher than we actually proved in Section 4.7. We do not know at the moment whether they are of the special feature of the test problem or they simply indicate that the convergence analysis can be improved.

89 10 1 t=1 o--Global x--Interior o

10 0

x o o o o o o o o

10 -1

x

x

x x

x

1

x x

x

10 -2 10 -2

10 -1

10 0

Figure 4.1: Errors in H 1 Norm for 1 10 1 t=1 o--Global x--Interior 10 0

o x o o

x

o x

o o o

10 -1

o o

x

1

x x

x x

x

10 -2 10 -2

10 -1

Figure 4.2: Errors in H 1 Norm for 2

10 0

90 10 1 t=0.0001 o--Global x--Interior

o

0.5

10 0

x o

o o

o

o o

o

o

x

x

1

x x x x x

10 -1

x

10 -2 10 -2

10 -1

10 0

Figure 4.3: Errors in H 1 Norm for 1 10 1 t=0.0001 o--Global x--Interior o

10 0

x o o o o o o

10 -1

o o

x

x

x

1

x x

x x

x

10 -2 10 -2

10 -1

Figure 4.4: Errors in H 1 Norm for 2

10 0

91 10 0 o

t=1 o--Global x--Interior

x

o o o

x

o o

10 -1

x

o

1

x

o

x

o

x x x

x

10 -2 10 -2

10 -1

10 0

Figure 4.5: Errors in H 1 Norm for w 10 0

t=0.0001 o--Global x--Interior

o

x

o o

10 -1

x

o o

x

o

1

x

o x

o x

o x x x

10 -2 10 -2

10 -1

Figure 4.6: Errors in H 1 Norm for w

10 0

92 10 0 t=1 o--Global o

x--Interior 10 -1

x o o

x

o

10 -2 o

10 -3

x x

x

o o

2

x

o o

x x

x

10 -4 10 -2

10 -1

10 0

Figure 4.7: Errors in L2 Norm for 1 10 0 t=1 o--Global o

x--Interior 10 -1

x o o

x

o o o

10 -3

x x

x

o o

x

2

o

10 -2

x x

x

10 -4 10 -2

10 -1

Figure 4.8: Errors in L2 Norm for 2

10 0

93 10 0 t=0.0001 o--Global x--Interior

o x

10 -1 o

1.5 o

x

2

o x

o

10 -2

o x

o x

o o

x x

10 -3

x x

10 -4 10 -2

10 -1

10 0

Figure 4.9: Errors in L2 Norm for 1 10 0 t=0.0001 o--Global x--Interior

o x

10 -1

o x o

10 -2 o o o o

10 -3

o o

x

x

2

x

x

x

x x

10 -4 10 -2

10 -1

Figure 4.10: Errors in L2 Norm for 2

10 0

94 10 -1

o

t=1

x

o--Global o

x--Interior o

x

10 -2 o x

2

o o x

o x

o

10 -3

o

x

x x

x

10 -4 10 -2

10 -1

10 0

Figure 4.11: Errors in L2 Norm for w 10 -1 t=0.0001 o

o--Global

x

x--Interior o

10 -2

x o x

2

o o

x

o x

10 -3 o o

x

x

o x x

10 -4 10 -2

10 -1

Figure 4.12: Errors in L2 Norm for w

10 0

95

APPENDIX A AN APPROXIMATION RESULT 5.1 Introduction The purpose of this appendix is to prove Theorem 5.3.1, which is due to Arnold 7]. This approximation result was used extensively in Chapter 4 (as Theorem 4.7.1) for proving the global convergence of the Arnold-Falk element for the Reissner-Mindlin plate model under the simply supported boundary condition. Recall that Ph is the space of continuous piecewise linear functions. We shall start with a result by Scott and Zhang 39].

Theorem 5.1.1. Assume that  is a convex polygon. Let ; = @  and Ph; =

f vj; : v 2 Ph g  H 1(;). There exists a projection Ih : H 1 ! Ph such that if uj; 2 Ph; then Ih uj; = uj;. Moreover ku ; Ih uks  Chl;skukl for 0  s  l  2 l > 1=2: (5.1.1) Using this, we can quickly prove:

Lemma 5.1.2. Let w 2 H 1 be a function for which wj; 2 Ph;. Dene wh 2 Ph by Z

Z



grad wh  grad v = grad w  grad v for all v 2 P h wh = w on ;: 

Then

kw ; whk1  C

inf

2Ph =w on ;

kw ; k1

kw ; wh k0  Chkw ; whk1 kw ; whks  Cht;skwkt s = 0 1

(5.1.2) (5.1.3)

t = 1 2:

(5.1.4)

96

Proof. The rst two estimates are completely standard. We take  to be the interpolant of Theorem 5.1.1 to get the third.

The outline of this chapter is as follows. Section 5.2 constructs the approximation operator and section 5.3 proves that it has the desired property.

5.2 The Construction of the Approximation Operator In this section we study a nite element method for the nonhomogeneous Dirichlet problem for the Poisson equation. We will prove some of the properties of the nite element method here and we will show in the next section that the approximation operator determined by the nite element solution is the one we need. For simplicity, we will use notation jjt to denote kkt @ in this chapter (instead of its old meaning as the semi-norm on H t ).

Lemma 5.2.1. Given p 2 H 1, let g = pj; and let gh be the L2(;)-projection of g into Ph;. Dene ph 2 Ph by Z

Z



grad ph  grad q = grad p  grad q for all q 2 P h ph = gh on ;: (5.2.1)

Then



kp ; phks  Cht;skpkt

0s1

1  t  2:

Proof. Dene ph 2 H 1 by

 ph =  p in 

ph = gh on ;:

Since p ; ph is harmonic, we have

kp ; ph k0  C jg ; gh j;1=2

kp ; ph k1  C jg ; ghj1=2 :

(5.2.2)

97

Now using a standard duality argument and standard approximation results for the L2-projection into Ph; together with the trace theorem we get

jg ; ghj;1=2 = f 2Hsup (;) hg j;f jg1h=2 f i 1=2

hg ; gh f ; f I i jf j1=2 f 2H (;)  Ch1=2jg ; ghj0  Chjgj1=2  Chkpk1 =

sup

1=2

(5.2.3)

where f I is the L2(;) projection of f on Ph;. Although ; is not suciently smooth to dene the space H 3=2 (;) intrinsically, we can dene H 3=2 (;) to be the space of functions in H 1(;) whose restrictions to each edge e of the polygon belong to H 3=2 (e), and use as the norm

v3=2 :=



X

e2@ 

kvk2H

!1=2 3=2

(e)

:

Then

jg ; ghj;1=2  Ch1=2jg ; gh j0  Ch2jgj3=2  Ch2kpk2: (5.2.4) From jg ; ghj0  C jgj0 and the inverse inequality we can obtain jg ; gh j1  C jgj1.

Then, by the interpolation theorem we get

jg ; ghj1=2  C jgj1=2  C kpk1 jg ; ghj1=2  Chjgj3=2  Chkpk2:

(5.2.5)

Combining (5.2.2){(5.2.5) we get

kp ; phks  Cht;skpkt

s = 0 1 t = 1 2:

(5.2.6)

Now Z



Z

grad ph  grad q = grad ph  grad q for all q 2 P h ph = ph on ; 

98

Then, using (5.1.4) in the case t = 1 we obtain

kph ; ph ks  Ch1;skph k1

s = 0 1:

Thus combining the above with (5.2.6) in the case t = 1 we get

kp ; phks  Ch1;s(kpk1 + kph k1)  Ch1;skpk1

s=0 1

where in the last step we use (5.2.6) for s = 1 and t = 1. Now let Ih p be the usual piecewise linear interpolant of p so that egh := Ih pj; is the piecewise linear interpolant of g, and dene peh 2 Ph by Z



Z

grad peh  grad q = grad p  grad q for all q 2 P h peh = geh on ;: 

Then

kp ; pehk1  C

inf

2Ph =egh on ;

kp ; k1  C kp ; Ihpk1  Chkpk2:

(5.2.7)

Next, dene w 2 H 1 by

w = egh ; gh on ;:

 w = 0 in  Note that peh ; ph 2 Ph, and Z



Z

grad(peh ; ph )  grad q = grad w  grad q = 0 for all q 2 P h 

peh ; ph = w on ;:

Then by the Lemma 5.1.2, we have kpeh ; ph ks  Ch1;skwk1, for s = 0, 1. Since

kwk1  C jegh ; ghj1=2  C (jg ; egh j1=2 + jg ; ghj1=2)  Chjgj3=2  Chkpk2

99

we get

kpeh ; ph ks  Ch2;skpk2 which, together with (5.2.7) gives kp ; ph k1  Chkpk2.

Finally we use duality to prove that kp;ph k0  Chkp;phk1, and thus kp;phk0  Chkpk2. Namely, we dene z by

;  z = p ; ph

in 

z = 0 on ;:

Then kzk2  C kp ; ph k0, and

kp ; k ; ph 20 =

Z



(p ; ph )  z =

Z



Z

@z grad(p ; ph )  grad z ; (g ; gh ) @n

 kp ; phk1 inf kz ; k1 + jg ; ghj 2P

  @z ;1=2  @n

h

 Chkp ; phk1 kzk2 + Ch2jgj3=2kzk2  Ch2kpkkp ; ph k0

   

;

T

T 1=2

as desired. This completes the proof for s = 0 and 1, and t = 1 and 2. The extension to real indices follows by interpolation.

5.3 The Main Result For p 2 H 1 , let h p = ph be the nite element solution dened in Lemma 5.2.1. In the following, we shall prove that h is what we need. To do so, we should keep in mind two important properties of h : equation (5.2.1) and that h preserves Ph; on the boundary.

100

Theorem 5.3.1. Assume that u 2 H 2 , where  is a convex polygon. Then the operator h : H 1 ! Ph constructed in Lemma 5.2.1 satises

kp ; h pk0  Ch1=2+kpk1=2+ kp ; h pk1  Ch1=2kpk3=2 jp ; h pj;1=2  Ch1=2+kpk1=2+

(5.3.1) (5.3.2) (5.3.3)

for any 0 <   1=2. Here C is independent of  and C depends on , but not h. Proof. Inequality (5.3.2) is already proved in Lemma 5.2.1. Inequality (5.3.3) is also straightforward: since h p is the L2(;) projection of p on Ph;, we have

jp ; hpj;1=2  Ch1=2jp ; h pj0  Ch1=2+jpj  Ch1=2+kpk1=2+ where we use the trace theorem in the last step. We now prove (5.3.1) in three steps: rst for p 2 H 1 , then for p such that pj; 2 Ph;, and nally for all p 2 H 1 . Using an inverse inequality 43,Theorem 3.1], we obtain

kzhk3=2;  Ch;1=2+kzh k1 for all zh 2 Ph which implies that zh 2 H 3=2; . For all T 2 Th, applying inequality 32] kuk1+s T  C ( kuk2 T + ;s=(1;s)kuk1 T ) for u = z ; h z, = h1=2;, and s = 1=2 ;  yields kz ; hzk3=2; T  C(h1=2;kz ; h zk2 T + h;1=2;kz ; h zk1 T ) for all z 2 H 2 : Summing up inequalities of above type for all T 2  and noting that the second order derivative of h z vanishes, we obtain

kz ; hzk3=2;  C(h1=2;kzk2 + h;1=2;kz ; h zk1)

for all z 2 H 2 ):

101

Then applying Lemma 5.2.1 for s = 1 and t = 2 yields

kz ; h zk3=2;  Ch1=2;kzk2 for all z 2 H 2: Now if p 2 H 1 then h p 2 P h, so if both p, z 2 H 1 ,

(5.3.4)

(grad h p grad z) = (grad h p grad h z) = (grad p grad h z): For a given p 2 H 1 , we will use a duality argument to get (5.3.1). Taking z 2 H 2 \H 1 with ;  z = p ; h p and kzk2  C kp ; h pk0 for h z as described in Lemma 5.2.1, we get

kp ; hpk20 = (grad(p ; h p) grad z) = (grad p grad(z ; h z))  k grad pk;1=2k grad(z ; hz)k1=2;  kpk1=2+kz ; h zk3=2;  Ch1=2+kpk1=2+kzk2  Ch1=2+kpk1=2+kp ; h pk0 which proves (5.3.1) for p 2 H 1. Assume p 2 H 1 has the property that pj; 2 Ph;. Let Ih denote the Scott-Zhang interpolant 39]. Then, since Ih p = p on ; and, using what we just proved and the fact that Ih is bounded in H 1=2+, we obtain

kp ; hpk0 = k(p ; Ihp) ; h (p ; Ihp)k0  Ch1=2+kp ; Ihpk1=2+  Ch1=2+kpk1=2+: This completes the proof of the second case. Finally for the general case of p 2 H 1 we use the same decomposition as in the proof of Lemma 5.2.1. Namely we dene ph 2 H 1 by  ph =  p in  ph = gh on @  where gh is the L2(;)-projection of g = pj; into Ph;. Then

kp ; phk0  jg ; gh j;1=2  Ch1=2+jgj  Ch1=2+kpk1=2+:

102

Also

kp ; phk1=2+  jg ; ghj  Cjgj  Ckpk1=2+:

Here we have used an inverse inequality to obtain that the L2(@ )-projection is bounded in H  (@ ). We thus have

kph k1=2+  Ckpk1=2+: Finally we have h ph = h p, so

kph ; h pk0 = kph ; h ph k0  Ch1=2+kph k1=2+  Ch1=2+kpk1=2+: This completes the proof.

103

APPENDIX B A REGULARITY RESULT The purpose of this Appendix is to prove Theorem 6.1 on the regularity of the exact solution of the singularly perturbed Stokes-like system under the soft simply supported boundary condition. This is done assuming that the domain  is smooth. So far, we cannot prove the same result for a convex polygonal domain.

Theorem 6.1. Let  denote a smooth domain, and let F 2 H 1 and K 2 H 2 \ L^ 2 . Then there exists a unique solution ( P ) 2 H 2 H 1 \ L^ 2 to the equations (C E () E ()) ; ( curl P ) = (F ) for all  2 H 1

;( curl Q) ; ;1t2 (curl Q curl P ) = (K Q)

for all Q 2 H 1 :

(6.1) (6.2)

Moreover,

kk2 + kP k1 + tkP k2  C ; t;1=2 (kF k;1=2 + kK k1=2) + t(kF k1 + kK k2)
kP k1=2  C ; kF k;1=2 + kK k1=2 + t(kF k1=2 + kK k3=2)
kk3=2 + tkP k3=2  C ; kF k;1=2 + kK k1=2 + t3=2(kF k1 + kK k2)
: Proof. We rst dene some notations. Let 



Mn := n  C E ( )n = D @@n  n +  @@s  s   D (1 ;  ) @ @ Ms := s  C E ( )n = 2 @n  s +  @n  n

(6.3) (6.4) (6.5)

104

on @ , where s and n are the unit tangential and outward normal directions, respectively. Then consider a reduced problem: Find (0 P0) 2 H 2 H 1 \ L^ 2 such that

; div C E (0) ; curl P0 = F ; rot 0 = K

(6.6) (6.7)

together with boundary conditions

Mn 0 = 0

0  s = 0:

By the standard theory on the elliptic system, we have

k0ks+1 + kP0 ks  C (kF ks;1 + kK ks) Now set

E =  ;  0

for all real s  0:

(6.8)

P E = P ; P0 :

In the light of (6.8), we need only estimate E and P E . Actually, we have the following theorem.

Theorem 6.2. Under the same conditions of Theorem 6.1, there exists a constant C depending only on the domain  such that

kE k1 + kP E k0 + tkP E k1  C ; t1=2(kF k;1=2 + kK k1=2 ) + t3=2(kF k1=2 + kK k3=2)
kE k2 + tkP E k2  C ; t;1=2(kF k;1=2 + kK k1=2 ) + t(kF k1 + kK k2)
: We claim the above is enough for our purpose.

(6.9) (6.10)

105

Proof of Theorem 6.1. Suppose momentarily that Theorem 6.2 is proved. Then estimate (6.3) can be obtained by combining (6.9) (for kP E k1), (6.10), and (6.8). Moreover,

kE k23=2  C kE k1kE k2  C ; t1=2(kF k;1=2 + kK k1=2) + t3=2(kF k1=2 + kK k3=2)
 ; t;1=2(kF k;1=2 + kK k1=2) + t(kF k1 + kK k2)
 C (kF k2;1=2 + kK k21=2 + t2kF k21=2 + t2kK k23=2 + t3kF k21 + t3kK k2
 C ; kF k2;1=2 + kK k21=2 + t3kF k21 + t3 kK k22
where we use the fact that

kF k1=2  C (t1=2kF k1 + t;1kF k;1=2) So

kK k3=2  C (t1=2kK k2 + t;1kK k1=2 ):

kE k3=2  C ; kF k;1=2 + kK k1=2 + t3=2 (kF k1 + kK k2)
:

Similarly

kP E k1=2  C ; kF k;1=2 + kK k1=2 + t(kF k1=2 + kK k3=2)
and

kP E k3=2  C ; t;1(kF k;1=2 + kK k1=2) + t1=2(kF k1 + kK k2)
: Combining these estimates on E and P E with the estimates in (6.8) for 0 and P0 then gives (6.4) and (6.5).

Therefore it remains to prove Theorem 6.2. From the denitions we get (C E (E ) E ()) ; ( curl P E ) = ;hMs0   si for all  2 H 1

;(E + ;1t2 curl P E curl Q) = ;1t2(curl P0 curl Q)

(6.11)

for all Q 2 H 1 : (6.12)

106

We will prove Theorem 6.2 by choosing the appropriate test functions in these equations. First we need a lemma.

Lemma 6.3. Under the same conditions of Theorem 6.1, there is a constant C such that for r 2 H 1 (@ )

jhE  s rij Ct3=2(krk0 @ + tkrk1 @)(kF k0 + kK k1 + kP E k1) + Ct1=2krk0 @ kE k1 : Proof of Lemma 6.3. We dene the usual boundary-tted coordinates in a neighborhood of the boundary. Let 0 be a positive number less than the minimum radius of curvature of @  and dene

0 = f z ; nz jz 2 @  0 <  < 0 g where nz is the outward unit normal to  at z. Let z() = (X () Y ()),  2 0 L), be a parametrization of @  by arclength which we extend L;periodically to  2 R. The correspondence ( ) ! z ; nz = (X () ; Y 0() Y () + X 0()) is a dieomorphism of (0 0 ) R=L on 0. For any function f , let f^( ) denote the change of variable to the ( )-coordinate. Now, we dene an extension R of r to 0 by

R( ) = r^()e; =t : Then nd a smooth cut-o function  which is a function of  alone, independent of  and t, and identically one for 0    0=3, identically zero for  > 20=3. Thus R gives an extension to all of  and, by simple computations,

kRk0  Ct1=2krk0 @ kRk1  C (t;1=2 krk0 @ + t1=2 krk1 @ ):

107

Using integration by parts and (6.12) with Q = R we obtain

hE  s ri = (curl(R) E ) ; (R rot E ) = ;;1t2(curl(P E + P0 ) curl(R)) ; (R rot E ) Applying the Schwartz inequality and the estimates on R leads to the proof of the lemma. We are now in the position to prove Theorem 6.2. Proof of Theorem 6.2. Taking  = E in (6.11) and Q = P E in (6.12) gives

(C E (E ) E (E )) + ;1t2 (curl P E curl P E )

= ;hMs 0 E  si ; ;1t2(curl P0 curl P E ):

(6.13)

We bound the rst term on the right hand side using Lemma 6.3 and the bounds (6.8) on 0 :

jhMs0 E  sij  C (t1=2kMs0 k0 @ + t3=2 kMs0k1 @)(tkF k0 + tkK k1 + tkP E k1) + Ct1=2kMs 0k0 @ kE k1  C ; t1=2(kF k;1=2 + kK k1=2) + t3=2(kF k1=2 + kK k3=2 )
 ; tkF k0 + tkK k1 + tkP E k1
+ Ct1=2(kF k;1=2 + kK k1=2)kE k1:

(6.14)

For the second term on the right hand side of (6.13) we have

jt2(curl P0 curl P E )j  Ct(kF k0 + kK k1) tkP E k1:

(6.15)

Next we choose Q with zero average and curl Q = P E , the L2-projection onto the rigid motions (the space spanned by f (a ; by c + bx)ja b c 2 R g) in (6.12), to get (E P E ) = ;;1 t2(curl(P E + P0) P E )

108

which implies

kP E k0  Ct(tk curl P E k0 + tkF k0 + tkK k1): Combining (6.13){(6.16) and using the equivalence between j(C E ( ) E ( ))j1=2 + kP k0 gives

(6.16)

k k1

and

kE k1 + tkP E k1  C ; t1=2(kF k;1=2 + kK k1=2) + t3=2(kF k1=2 + kK k3=2)
where we use the fact that

kF k0  C (t1=2kF k1=2 + t;1=2kF k;1=2)

kK k1  C (t1=2kK k3=2 + t;1=2kK k1=2 ):

Finally we choose  in (6.11) with rot  = P E to get Thus

  s = 0 on @ 

kk1  C kP E k0

(P E P E ) = (C E (E ) E ()):

kP E k0  C kE k1:

This completes the proof of the rst estimate of Theorem 6.2. To get the second estimate we use elliptic regularity. From (6.11) we see

; div C E (E ) = curl P E in  MsE = ;Ms0 MnE = 0 on @ : Therefore

kE k2  C ; kP E k1 + kMs0k1=2 @ + kP E k0
 C ; t;1=2(kF k;1=2 + kK k1=2) + t1=2(kF k1=2 + kK k3=2) + kF k0 + kK k1
 C ; (t;1=2 (kF k;1=2 + kK k1=2) + t1=2(kF k1=2 + kK k3=2)


109

as desired. Similarly, by (6.12)

;  P E =  P0 ; t;2 rot E @P E = ; @P0 ; t;2E  s @n @n

in  on @ 

so

kP E k2  C (kP0k2 + t;2 kE k1)  C ; kF k1 + kK k2 + t;3=2 (kF k;1=2 + kK k1=2)
+ t;1=2 (kF k1=2 + kK k3=2) which is the desired estimate on kP E k2. Combining the standard interpolation theory and Theorem 6.1, we get

Corollary 6.4. Under the same conditions of Theorem 6.1, we have

kk3=2+ + kP k1=2+ + tkP k3=2+  Ct;; kF k0 + t3=2kF k1
kk3=2+ + kP k1=2+ + tkP k3=2+  Ct;kK k2 for 0    1=2.

for K = 0 (6.17)

for F = 0

(6.18)

110

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