Regularity estimates for elliptic boundary value ... - Semantic Scholar
J. Numer. Math., Vol. 11, No. 2, pp. 75–94 (2003) ° c VSP 2003
Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains C. BACUTA ¤ , J. H. BRAMBLE †, and J. Xu¤ Received 10 December, 2002
Communicated by R. Lazarov
Abstract — We consider the model Dirichlet problem for Poisson’s equation on a plane polygonal convex domain W with data f in a space smoother than L2 . The regularity and the critical case of the problem depend on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates for the critical case. As a consequence, sharp error estimates for the corresponding discrete problem are proved. Some classical shift estimates are also proved using the powerful tools of interpolation theory and mutilevel approximation theory. The results can be extended to a large class of elliptic boundary value problems. Keywords: interpolation spaces, nite element method, multilevel decomposition, shift theorems, subspace interpolation
1. INTRODUCTION Regularity estimates of the solutions of elliptic boundary value problems in terms of Sobolev norms of fractional order are known as shift theorems or shift estimates. The shift estimates for the Laplace operator with Dirichlet boundary conditions with non-smooth data on polygonal domains are well known (see, e.g, [2,21,23,27]). The classical regularity estimate for the case when W is a convex polygonal domain in R2 , with boundary ¶ W, is as follows: If u is the variational solution of ½ ¡ Du = f in W (1.1) u = 0 on ¶ W then, u 2 H 2 (W) and kukH 2 6 Ck f kH 0
8 f 2 H 0(W) = L2 (W):
(1.2)
Let w be the radian measure of the largest corner of W (w < p ), and let s0 = minf1; p =w ¡ 1g. If u is the variational solution of (1.1), then for 0 < s < s0 , it ¤ Dept. of Mathematics, The Pennsylvania State University University Park, PA 16802, USA † Dept. of Mathematics, Texas A & M University, College Station, TX 77843, USA The work of the second author was supported under NSF Grant No. DMS-9973328. The work of the rst and the third authors was supported under NSF Grant No.DMS-0209497.
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is known (see e.g. [16]) that kukH 2+s 6 C(s)k f kH s
8 f 2 H s (W):
(1.3)
Here, H s(W) is the interpolation space between H1 (W) and L2 (W) s¡ distance from L2 (W). In this paper we will prove a stronger version of (1.3) and approach the problem in the critical value situation s = s0 . We prove that, for s0 < 1, the following estimate holds. kukB2+s0 (W) 6 ck f kBs0 (W) 8 f 2 Bs10 (W) (1.4) ¥
0 B2+s ¥ (W)
1
Bs10 (W)
where and are standard interpolation spaces de ned in Section 3. It is worth noting here that the space Bs10 (W) contains all the Hilbert spaces Hs with s > s0 . The estimate (1.4) leads to new nite element convergence estimates. The method presented in this paper can be extended to other boundary conditions such as Neumann or mixed Neumann–Dirichlet conditions. The technique involved in proving shift results is the real method of interpolation of Lions and Peetre ([3,24,25]), combined with multilevel approximation theory. The cases q = 2, q = 1 and q = ¥, where q is the second index of interpolation, are of special importance for our problem. The following type of interpolation problem is essential for our approach. If X and Y are Sobolev spaces of integer order and XK is a subspace of codimension one of X , then how can one characterize the interpolation spaces between XK and Y for q = 2 and q = ¥? The problem was studied in [20,21] and [1] for q = 2 and particular spaces X and XK . This paper gives an interpolation result of this type for the case q = ¥ . The remaining part of the paper is structured as follows. The interpolation results presented in Section 2 give a formulas for the norms on the intermediate subspaces [XK ;Y ]s;2 and [XK ;Y ]s;¥ when XK is of codimension one. In Section 3 the main result concerning the codimension-one subspace interpolation problem is presented. Shift theorems for the Poisson equation on polygonal domains are considered in Section 4. In the last section, a straightforward application of the interpolation results is shown to lead to some new estimates for nite element approximations. 2. INTERPOLATION RESULTS In this section we give some de nitions and results concerning interpolation spaces via real method of interpolation of Lions and Peetre (see [3,4,24]). 2.1. Interpolation between Banach spaces Let X ;Y be Banach spaces satisfying for some positive constant c, ½ X is a dense subset of Y and kukY 6 ckukX 8 u 2 X where kukX and kukY are the norms on X and Y , respectively.
(2.1)
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77
The interpolation spaces [X ;Y ]s;q , 1 6 q 6 ¥, and s 2 (0; 1) are de ned using the K function, where for u 2 Y and t > 0, K(t;u) := inf (ku0 k2X + t 2 ku ¡ u0 kY2 )1=2 : u0 2X
Then, for q < ¥, the space [X ;Y ]s;q consists of all u 2 Y such that Z ¥ 0
(t ¡ s K(t;u))q
dt 0
K(t;u)2 < ¥:
The norm on [X ;Y ]s;q is de ned by kuk2[X;Y ]s;q
:=
Z ¥ 0
(t ¡ s K(t; u))q
dt t
and the norm on [X ;Y ]s;¥ is de ned by kuk [X ;Y ]s;¥ := sup t ¡ s K(t;u): t>0
Remark 2.1. Since K(t; u) 6 tkukY for all u 2 Y , the interval (0;¥) in the above de nitions can be replaced by any subinterval (A; ¥). The new norms obtained are equivalent with the original norms. 2.2. Interpolation between Hilbert spaces An extended Hilbert interpolation theory can be found in [24]. For completeness and consistence of notation we present in this section the interpolation results used in this paper. Let X ;Y be separable Hilbert spaces with inner products (¢; ¢)X and (¢; ¢)Y , respectively, and satisfying (2.1). Let D(S) denote the subset of X consisting of all elements u such that (2.2) v ! (u; v)X ; v 2 X is continuous in the topology induced by Y . For any u in D(S) the anti-linear form (2.2) can be uniquely extended to a continuous anti-linear form on Y . Then by Riesz representation theorem, there exists an element Su in Y such that (u;v)X = (Su;v)Y 8 v 2 X : (2.3) In this way S is a well de ned operator with domain D(S) in Y . The next result illustrates the properties of S (see [24]).
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Proposition 2.1. The domain D(S) of the operator S is dense in X and consequently D(S) is dense in Y . The operator S : D(S) » Y ! Y is a bijective, selfadjoint and positive de nite operator. The inverse operator S¡ 1 : Y ! D(S) » Y is a bounded symmetric positive de nite operator and (S ¡ 1 z; u)X = (z; u)Y
(2.4)
8 z 2 y; u 2 X :
The next lemma provides the relation between K(t;u) and the connecting operator S. A proof of the lemma is given in [2]. Lemma 2.1. For all u 2 Y and t > 0 , ¡ K(t;u)2 = t 2 (I + t 2 S¡ 1 )¡
1
¢ u; u Y :
For the special case q = 2 (X ; Y Hilbert spaces), due to the spectral or multilevel representation of the norm on [X ;Y ]s;2 , the de nition of the norm is slightly changed as follows: Z ¥ kuk2[X ;Y ]s;2 := c2s t ¡ (2s+1)K(t;u)2 dt 0
where
cs :=
µZ
0
¥
t 1¡ 2s dt t2 + 1
¶¡
1=2
=
r
2 sin(p s): p
With the new de nition for the norm of [X ;Y ]s;2 it is natural (see the multilevel representation case in Section 3) to de ne [X ;Y ]0;2 := X ;
[X ;Y ]1;2 := Y:
Remark 2.2. Lemma 2.1 yields other expressions for the norms on [X ;Y ]s;2 and [X ;Y ]s;¥ . Namely, 2
kuk and
[X;Y ]s;2
=
c2s
Z ¥ 0
t¡
2s+1
kuk2 [X;Y ]s;¥ = sup t 2(1¡ t>0
s)
¡
(I + t 2 S ¡ 1 )¡ 1 u;u
¡
(I + t 2 S¡
1 ¡ 1
¢
) u;u
Y
¢
Y
dt
(2.5)
:
(2.6)
2.3. Interpolation between subspaces of a Hilbert space Let k = spanfj g be a one-dimensional subspace of X and let Xk be the orthogonal complement of k in X in the (¢; ¢)X inner product. We are interested in determining the interpolation spaces of Xk and Y , where on Xk we consider again the (¢; ¢)X inner product. To apply the interpolation results from the previous section we need to check that the density part of the condition (2.1) is satis ed for the pair (Xk ;Y ).
New regularity estimates
79
For j 2 k , de ne the linear functional Lj : X ! C, by Lj u := (u; j )X ;
u 2 X:
The following result is an extension of Kellogg’s lemma [21]. The proof can be found in [1]. Lemma 2.2. Let k be a closed subspace of X and let Xk be the orthogonal complement of k in X in the (¢; ¢)X inner product. The space Xk is dense in Y if and only if the following condition is satis ed: ½ Lj is not bounded in the topology of Y (2.7) for all j 2 k ; j 6= 0: For the remaining part of this section we assume that Lj is not bounded in the topology of Y, so the condition (2.1) is satis ed for the pair (Xk ;Y ). We denote Xk by Xj . It follows from the previous section that the operator Sj : D(Sj ) » Y ! Y de ned by (u; v)X = (Sj u;v)Y 8 v 2 Xj (2.8) has the same properties as S. Consequently, the norms on the intermediate spaces [Xj ;Y ]s;2 and [Xj ;Y ]s;¥ are given by: 2
kuk and
[Xj ;Y ]s;2
=
c2s
Z ¥ 0
t¡
2s+1
kuk2 [Xj ;Y ]s;¥ := sup t 2(1¡
¡
s)
t>0
¢ (I + t 2 S¡j 1 )¡ 1 u; u Y dt
¡
(I + t 2 Sj¡ 1 )¡ 1 u;u
¢
Y
(2.9)
(2.10)
:
Our aim in this section is to determine suf cient conditions for j such that the norm [Xj ;Y ]s;q (for q = 2 and q = ¥) can be compared with more familiar intermediate norms which are independent of j . First, we note that the operators Sj and S are related by the following identity: Sj¡ 1 = (I ¡ Qj )S¡
1
(2.11)
where Qj : X ! k is the orthogonal projection onto k = spanfj g. The proof of (2.11) follows easily from the de nitions of the operators involved. Next, (2.11) leads to a new formula for the norms on [Xj ;Y ]s;¥ and [X ;Y ]s;2 . Theorem 2.1. For any u 2 Y we have, kuk2 [Xj ;Y ]s;2 = kuk2 [X ;Y ]s;2 + c2s
Z ¥ 0
t¡
2 2s+3 j(u; j )Y;t j
(j ; j )X;t
dt
(2.12)
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80
and
Ã
kuk2[Xj ;Y ]s;¥ = sup t 2¡ t>0
2s
(u; u)Y;t + t 4¡
2 2s j(u; j )Y;t j (j ; j )X;t
!
:
(2.13)
dt
(2.14)
In particular for u 2 Xj we have kuk2 [Xj ;Y ]s;2 = kuk2 [X ;Y ]s;2 + c2s and
Ã
kuk2[Xj ;Y ]s;¥ = sup t 2¡ t>0
where
Z ¥ 0
t¡
(2s+1) j(u; j )X;t j
and
(j ; j )X;t
j(u; j )X;t j2 2s (u; u)Y;t + t ¡ 2s (j ; j )X;t
¡ ¢ (u; v)X;t := (I + t 2 S¡ 1 )¡ 1 u;v X ¡ ¢ (u; v)Y;t := (I + t 2 S¡ 1 )¡ 1 u; v Y
2
!
(2.15)
8 u;v 2 X
(2.16)
8 u;v 2 Y:
(2.17)
Proof. The rst two formulas follows immediately from Remark 2.2 and the following identity ¡
¢ j(u; j )Y;t j2 (I + t 2 S¡j 1 )¡ 1 u; u Y = (u; u)Y;t + t 2 : (j ; j )X;t
The proof of the identity is based on (2.11). A detailed proof of it can be found in [2]. Using (2.4) and a simple manipulation of the operator S we get t 2 (u; j )Y;t = (u; j )X ¡ (u; j )X;t which, for u 2 Xj leads to (2.14) and (2.15).
¤
Theorem 2.1 can be easily extended to the case when K is of nite dimension(see [1]). Such an extension would be needed, for example, to treat the case of mixed Neumann–Dirichlet conditions or the case of the biharmonic problem. 3. MULTILEVEL REPRESENTATION OF INTERPOLATION SPACES Let W be a domain in R2 with boundary ¶ W. Assume that M1 » M2 » ¢ ¢¢ » Mk » : :: is a sequence of nite dimensional subspaces of H1 (W) whose union is dense in H 1 (W), and assume that an equivalent norm on H1 (W) is given by à !1=2 kuk1 :=
¥
å lk k(Qk ¡
k=1
Qk¡ 1 )uk2
(3.1)
New regularity estimates
81
where Qk denotes the L2 (W) orthogonal projection onto Mk, k¢k = k¢kL2 (W) , Q 0 = 0 and lk = 4k¡ 1 . The sequence fMk g can be taken for example the standard sequence of piecewise linear functions associated with a sequence of nested meshes. Proofs for the multilevel representation of the norm on H1 , for speci c choices of the spaces Mk can be found in [1,10,28,30]. 3.1. Scales of multilevel norms On H 1 (W) we consider the norm given by (3.1) and de ne H¡ 1 (W) to be the dual of H 1 (W). The elements of L2 (W) can be viewed as continuous linear functionals on H 1 (W) and we have the natural continuous and dense embeddings H 1 (W) » L2 (W) » H ¡ 1 (W): The projection Qk , k = 1; 2;: :: , can be extended to be de ned on H¡ 1 (W) by 8 u 2 H ¡ 1 (W); v 2 Mk
(Qk u; v)L2 = (u; v)
where (¢; ¢) on the right hand side represents the duality between H¡ 1 (W) and H 1 (W). One can easily check that the induced inner product on He (W) is given by (u; v)e :=
¥
å lke (qk u; qk v)
8 u; v 2 H e (W); e = ¡ 1; 1
k=1
where qk = Qk ¡ Qk¡ 1 . Then the pair (H 1 (W);L2 (W)) satis es the condition (2.1) and the operator S associated with the pair is given by Su =
¥
å lk qk u
(3.2)
8 u 2 D(S):
k=1
Thus, for X = H 1(W) and Y = L2 (W), we have ¥
(u;v)Y;t =
lk
å lk + t 2 (qk u;qk v);
u; v 2 Y
k=1
and
¥
(u;v)X;t =
l2
å lk +k t 2 (qk u;qk v);
u; v 2 X :
k=1
For any s 2 (0; 1), q = 1 or q = ¥, let H s (W) := [H 1 (W); L2 (W)]1¡
s;2
;
H 2+s(W) := [H 3 (W); H 1 (W)]1¡
s;2
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and Bsq (W) := [H 1 (W);L2 (W)]1¡
3 1 B2+s q (W) := [H (W); H (W)]1¡
s;q ;
s;q :
By using (2.5) and (2.6), one can easily check that kuk2H s (W) = and
¥
å lkskqk uk2
(3.3)
k=1
Ã
! l k kuk2Bs¥ (W) = sup t 2s å kqk uk2 : 2 l + t k t>0 k=1 ¥
(3.4)
One can verify using the above formula that B0¥ (W) = L2 (W) and B1¥ (W) = H 1 (W). In addition we have t 2s = ss (1 ¡ s)1¡ s lk¡ 1+s sup 2 t>0 lk + t which leads to kuk2Bs¥ (W) 6 ss (1 ¡ s)1¡ s kuk2Hs (W)
8 u 2 H s(W)
(3.5)
a well known embedding property. Remark 3.1. If we assume that the sequence of subspaces fMk g is chosen so that the following approximation and inverse inequalities hold with a constant c independent of j: (Ap)
ku ¡ Q j uk 6 c 2¡ j kukH 1
(Inv)
kuk 6 c 2 j kukH 1
8 u 2 H 1 (W)
8u 2 Mj
then, by a well known result of the approximation theory, an equivalent norm on [H 1 ; L2 ]1¡ s;q is given by kf2s j kq j ukg j>1 klq ;
0 < s < 1; 1 6 q 6 ¥:
In particular, an equivalent norm on Bs1 := [H 1 ;L2 ]1¡ ¥
å 2s j kq j uk;
s;1
is given by
0 < s < 1:
j=1
One can verify now that we have the following norm-inequality: kukBs1 6 ce ¡ with c independent of e .
1=2
kukH s+e
8 u 2 H s+e
(3.6)
New regularity estimates
83
3.2. Subspace interpolation results Next, we focus our attention on a speci c case of subspace interpolation associated with the pair (X ;Y ), where X = H 1(W) and Y = L2 (W). For a xed s0 in the interval (0;1), let J0 = 1 ¡ s0 and j 2 H ¡ 1 (W). By the Riesz representation theorem there exists a function j 2 H 1(W) such that (v; j ) = (v; j )1 =
¥
å lk (qk j ;v)
8 v 2 H 1 (W):
k=1
Since qi q j = 0 for i 6= j, we deduce that in fact qk j = lk¡ 1 qk j :
(3.7)
Next, we assume that the function j satis es the following condition: (C) There exist two positive constants c1 ;c2 such that c1 lks0 6 kqk j k2 6 c2 lks0 ;
k = 1; 2;: :: :
Note that the above condition is equivalent to c1 lk¡
J0
J0
6 kqk j k21 6 c2 lk¡
;
k = 1; 2;: :: :
Lemma 3.1. Let j 2 H ¡ 1 (W) satisfy (C) and let j be the corresponding H1 representation. Then the following conditions are also satis ed. (C.0) Hj1 is dense in [H 1 ; L2 ]1¡
s
for s < s0 . Here, Hj1 is the kernel of j .
(C.1) There exist two positive constants c1 ;c2 such that c 1t ¡
2J0
6 (j ; j )X;t =
¥
lk2 å lk + t 2 k qk j k2 6 c2t ¡ k=1
2J0
;
t > 1:
Proof. The constants involved in this proof might change at different occurrences. To prove (C.0), by Lemma 2.2, it is enough to show that the functional u ! (u; j )1 = (u; j );
u 2 H 1 (W)
(3.8)
is not continuous in the topology induced by Hs (W), (s < s0 ). Let fun g be the sequence in H 1 (W) de ned by un :=
n
å lk¡ s
k=1
0
qk j :
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Then,
n
(un ; j )1 = (un ; j ) =
å lk¡ s kqk j k2 ! ¥
as n ! ¥:
0
k=1
On the other hand kun k2H s =
n
å lks¡
2s0
kqk j k2
k=1
is uniformly bounded. Therefore, the functional de ned in (3.8) is not continuous and (C.0) is proved. To estimate (j ; j )X;t we observe that kqk j k2 = lk¡ 1 kqk j k21 and that
lk1¡ J0 å lk + t 2 = t ¡ k=1 ¥
2J0
¥
å
k=1
(4k =t 2 )1¡ J0 : (4k =t 2 + 1)
Using the standard convergence criteria for sums via integrals, the last sum can be ¤ estimated below and above by constants which are independent of t. Next theorem is the main subspace interpolation result of the paper. Theorem 3.1. Let j 2 H ¡ 1 (W) satisfy (C). Then [Hj1 ;L2 ]1¡ and
s;2
= [H 1 ;L2 ]1¡
Bs10 := [H 1; L2 ]1¡
s;2
:= H s(W);
» [Hj1 ;L2 ]1¡
s0 ;1
s < s0
(3.9) (3.10)
s0 ;¥ :
Proof. Recall that L2 (W) = Y and H 1 (W) = X . In order to prove (3.9) it is enough (by the density property (C.0), (2.14) and Remark 2.1) to prove that I :=
Z ¥ 1
t¡
2 (2(1¡ s)+1)) j(u; j )X;t j
(j ; j )X;t
dt 6 ckuk2H s
8 u 2 Xj
s=2
for u 2 H s (W) denote u˜k := lk kqk uk and u˜ := fu˜k g. Then we have kukHs = kuk ˜ l2 : Here (¢; ¢) is simply the L2 (W) inner product. Then, we have ¡ (u; j )X;t = (I + t 2 S¡ 1 )¡
1
u; j
¢
X
¥
=
k=1
For u 2 Xj we have (u; j )X = 0. Then ¥
å (qk u; j )X = 0:
k=1
lk
å lk + t 2 (qk u; qk j )X :
(3.11)
New regularity estimates
Consequently,
85
¥
1 (q u;qk j ): 2 k l k=1 k + t
(u; j )X;t = ¡ t 2 å
Thus, using the Cauchy–Schwarz inequality and (C) we obtain the estimate
lks0 =2 kqk uk: 2 l k=1 k + t ¥
j(u; j )X;t j 6 c2 t 2 å
(3.12)
Now we are prepared to estimate the integral I. The constant c, to be used next, may have different values at different places in which it appears. Let s1 = s0 ¡ s. Then, by (C.1) and the estimate (3.12), we have !2 à Z ¥ ¥ l 1¡ s0 =2 kqk uk dt I6c t ¡ 3+2¡ 2s0 +4 å k 2 l 1 k=1 k + t à ! Z ¥ ¥ s0 =2 l l ( ) m n 6c kqm ukkqn uk dt t 3¡ 2s1 å 2 2 1 m;n=1 (lm + t )(ln + t ) ¥
=c
å
(lm ln )s0 =2 kqm uk kqn uk
m;n=1
Z ¥ 1
t 3¡ 2s1 dt: (lm + t 2 )(ln + t 2 )
Next, we use the formula Z ¥ 0
t 3¡ 2J 1 a1¡ J ¡ b1¡ dt = (a + t 2 )(b + t 2 ) a¡ b c2J
J
; 0 < J < 2; J 6= 1; a; b > 0: (3.13)
The integral can be calculated by elementary calculus methods. If a = b, then the right side of the above identity is replaced by 1¡c2J a¡ J . Thus, J
Z ¥ 1
t 3¡ 2s1 dt 6 (lm + t 2 )(ln + t 2 )
Z ¥ 0
l 1¡ s1 ¡ ln1¡ t 3¡ 2s1 dt = c¡s12 m 2 2 lm ¡ ln (lm + t )(ln + t )
Combining the above inequalities, we get I6c
¥
å
(lm ln )s1 =2
lm1¡
m;n=1
¡ ln1¡ lm ¡ ln s1
Let lmn = (lm ln )s1 =2
lm1¡
s1
lms=2 kqm uklns=2 kqn uk:
¡ ln1¡ lm ¡ ln s1
Then, the above estimate becomes I6c
¥
å
m;n=1
lmn u˜m u˜n :
s1
:
s1
:
C. Bacuta, J. H. Bramble, and J. Xu
86
An elementary calculation gives lmn =
n)(1¡ s1 ) ¡
2(m¡
2¡ n) ¡ 2¡
2(m¡
(m¡ n)(1¡ s1 ) (m¡ n)
jm¡ njs1
6 2¡
; m; n = 1;2; :: :
and by elementary estimates we obtain I 6 ckuk ˜ 2l2 = ck uk2H s which proves (3.9). Next, we prove (3.10). Let u 2 Bs10 . Then by (2.13) and Remark 2.1 we have that kuk 2[Xj ;Y ]1¡ Note that
s0 ;¥
Ã
6 c sup t 2s0 (u;u)Y;t + sup t 4¡ t>1
t>1
2 2J0 j(u; j )Y;t j (j ; j )X;t
!
:
sup t 2s0 (u; u)Y;t 6 sup t 2s0 (u; u)Y;t = kuk2Bs¥0 6 ckuk2Bs0 t>1
1
t>0
and, with the help of (C.1) we have sup t 4¡
2J0
t>1
j(u; j )Y;t j2 6 c sup t 4 j(u; j )Y;t j 2 : (j ; j )X;t t>1
(3.14)
For u 2 Y , we have (u; j )Y;t =
¥
¥
lk
1
å lk + t 2 (qk u;qk j ) = å lk + t 2 (qk u;qk j )
k=1
k=1
and by using condition (C), we obtain
lks0 =2 j(u; j )Y;t j 6 c2 å kqk uk: 2 l k=1 k + t ¥
The function
(3.15)
lks0 =2 kqk uk 2 l k=1 k + t ¥
t ! t 2 j(u; j )Y;t j = t 2 å
is an increasing function of t. As t ! ¥, the limit of the above function is exactly kukBs0 . Therefore, 1
and the proof is complete.
kuk2[Xj ;Y ]1¡
s0 ;¥
6 ckuk2[X ;Y ]1¡
s0 ;1
¤
New regularity estimates
87
4. A SHIFT THEOREM FOR THE LAPLACE OPERATOR ON CONVEX POLYGONAL DOMAINS In this section we will prove a stronger version of the estimate (1.3) and (1.4). 4.1. Shift estimates Let W be a convex polygonal domain in R2 , with boundary ¶ W and no right angles. Let V k (W) := H k (W) \ H01(W), k = 2 and k = 3. It is well known that for f 2 L2 (W) the variational problem has a unique solution u 2 V2 (W) and (1.2) holds. If W is an acute triangular domain then one can prove that for f 2 H1 (W) the solution of (1.1) belongs to V 3 (W) and (1.3) holds for s = 1. Using (1.2) and interpolation we obtain kukH 2+s 6 c(s)kuk[V 3 ;V 2 ]1¡
s;2
6 C(s)k f kH s
8 f 2 H s(W); 0 6 s 6 1:
Thus, without restricting the generality of the problem, we can assume that there exist one corner of measure w with p =2 < w < p . In fact, by a partition of unity type argument and using the regularity results for domains with smooth boundary, we can reduce the problem to the case when W is a domain with only one corner of measure w with p =2 < w < p . We will call this the ‘w -corner’ and we will assume that the vertex of the w -corner coincides with the origin of polar system of coordinates. Let a = p =w and s0 = a ¡ 1. Given f 2 H 1 (W), we consider the Dirichlet problem (1.1). The variational formulation of (1.1) is : Find u 2 H01 (W) such that Z
W
Ñu ¢ Ñv dx =
Z
W
f v dx
8 v 2 H01 (W):
(4.1)
Let z 2 d(W) be a cut-off function, which depends only on the distance r to the w -corner, is identically equal to one in a neighborhood of the w -corner and is identically equal to zero close to the part of ¶ W which does not contain the sides of the w -corner. Let y = j + uR , j (r; J ) = z r ¡ a sin(aJ ) and uR 2 H01(W) be the variatonal solution of (1.1) with f = Dj . One can check without dif culty that y 2 H ¡ 1 (W). Let Hy1 be de ned as the kernel of y as linear functional on H1 . Then, (see e.g., Theorem 9.8 in [16]) for f 2 Hy1 the (variational) solution of (1.1) Belongs to V 3 (W) and kukH 3 (W) 6 ck f kH 1 (W)
8 u 2 Hy1 (W):
(4.2)
Thus, by interpolation, we have kuk[V 3 (W);V 2 (W)]1¡ for all f 2 [H 1 (W)y ;L2 (W)]1¡
s;q .
s;q
6 ck f k[H 1 (W)y ;L2 (W)]1¡
s;q
Then, we have the following theorem.
(4.3)
C. Bacuta, J. H. Bramble, and J. Xu
88
Theorem 4.1. Let W be a convex polygonal domain in R2 with no right angles and with w the measure of the largest angle and let s0 = minf1; p =w ¡ 1g. If u is the variational solution of (1.1) then, for 0 < s < s0 there exist positive constant c(s) and C(s) such that kukH2+s 6 c(s)kuk[V 3 ;V 2 ]1¡
s;2
6 C(s)k f kH s
8 f 2 H s (W)
(4.4)
and for 0 < s0 < 1, kukB2+s0 (W) 6 c(s0 )kuk[V 3 ;V 2 ]1¡ ¥
s0 ;¥
6 C(s0 )k f kBs0 (W) 1
8 f 2 Bs10 (W):
(4.5)
Proof. Use (4.3) and apply Theorem 3.1 with j = y . The proof that (C) is satis ed is given later. The lower part of (4.4) or (4.5) follows by comparing the K ¤ functions associated with the two intermediate spaces. 4.2. Proving condition (C) Let W be a polygonal convex domain with the only one vertex O of measure w with p =2 < w < p and the remaining vertices denoted by S1 ;S2 ; :: : Sn . Let W=
n [
ti
i=1
where, for i = 1; :: : ;n, ti is a triangular domain with vertices Si ; O; Si+1 and O is taken to be the origin of a Cartesian system of coordinates in the plane. For i = 1; :: : ; n+ 1; let Gi denote the segment [O;Si ]. We assume, without loss of generality, that S1 lies on the positive semi-axis (see Fig. 1, the case n = 2). Let t1 = ft1 ;: :: ; tn g be the initial triangulation of W. We de ne multilevel triangulations recursively. For k > 1, the triangulation tk is obtained from tk¡ 1 by splitting each triangle in tk¡ 1 into four triangles by connecting the midpoints of the edges. The space Mk is de ned to be the space of all functions which are piecewise linear with respect to tk , vanish on ¶ W and are continuous on W. Let Qk denote the L2 (W) orthogonal projection onto Mk . We verify that the function y satis es the condition (C). To begin with, we will prove that the function j satis es (C). First we will prove that there exist a function wh 2 Mk which is supported in a ball of radius H = 1=2k¡ 1 = 2h centered at origin and wh is orthogonal on the space Mk¡ 1 . Let j1 ; :: : ; j 7 be the nodal functions in Mk¡ 1 corresponding to the nodal points in tk¡ 1 marked by ‘¯’ in the gure. Next, we consider the eight nodal functions j1 ;: :: j8 corresponding to the nodes marked by ‘?’). We de ne wh to be a linear combination wh of j1 ;: :: ; j8 , with coef cients independent of h such that (wh ; j j ) = 0; j = 1;: :: ;7; Hence, wh is orthogonal on the space Mk¡
1
(wh ; j ) 6= 0:
and consequently qk wh = wh .
New regularity estimates
G3
89
G2 ¯
¯ ¤
¯
¤
¤ ¯ ¤
¤ O
¤ ¯
¤
¯
¤ ¯
G1
Figure 1. The w-corner of W.
Then, (qk j ;v) (j ;qk v) j(j ; qk wh )j j(j ;wh )j > = sup = : kwh k kwh k v2L2 (W) kvk v2L2 (W) kvk
kqk j k = sup Note further that
kwh k 6 ch with c independent of h and by making the change of variable x = hxˆ in the integral which de nes the inner product (j ; wh ) we get j(j ; wh )j > ch2¡
a
for another constant c. Combining the above estimates we conclude that the lower part of the condition (C) holds. 1=2 For the upper part of (C) we rst note that kqk j k0 = lk kqk j k¡ 1 . To estimate kqk j k¡ 1 we let h = hh to be a cut off function which depends only on r and satis es
hh (r) = 1 for r 6 h=2;
hh (r) = 0 for r > h
jhh0 (r)j 6 c=h; jhh00 (r)j 6 c=h2
8 h=2 6 r 6 h
for some positive constant c. Then, kqk j k¡
1
= sup v2H 1 (W)
(qk j;v) kvk1
6
sup v2H 1 (W)
(qk (hj);v) kvk1
= M1 + M2 :
+ sup v2H 1 (W)
(qk ((1¡ h)j);v) kvk1
C. Bacuta, J. H. Bramble, and J. Xu
90
Using polar coordinates we have Z w µZ h hj ( ; qk v) = r¡ 0
a+1
0
¶ h qk v dr sin(aJ ) dJ :
Next, we integrate by parts with respect to the r variable (write r1¡ a as the derivative of r 2¡ a =(2 ¡ a )). Using the Cauchy–Schwarz inequality and the estimate for h0 we get ¡ ¢ j(hj ; qk v)j 6 c h¡ a+1 kqk vk + h¡ a+2 k(qk v)r k : The L2 and H 1 -stability of the L2 projection give kqk vk 6 chkvk1 ;
kqk vk1 6 ckvk1
with c independent of h (or k). Thus, j(hj ; qk v)j 6 ch2¡
a
M1 6 ch2¡
kvk1 ;
a
:
To estimate M2 , we observe rst that (1 ¡ hh )j 2 H 2 (W). Let Ph : H 2 (W) ! Mk be the interpolant associated with th = Tk . By applying standard approximation properties and (1.2) we obtain M2 = kqk (1 ¡ hh )j k¡
1
6 hkqk (1 ¡ hh )j k 6 hk(I ¡ Qk¡ 3
1 )(1 ¡ 3
h )j k
6 chk(I ¡ Ph )(1 ¡ hh )j k 6 ch k(1 ¡ hh )j kH 2 (W) 6 ch kD(1 ¡ hh )j )kL2 (W) : Using a simple computation in polar coordinates, and the estimates for the derivative of hh , we get kD(hh j )kL2 (W) 6 ch¡ 1¡ a : Combining the above inequalities, we have that M2 6 ch2¡
a
:
Hence, the upper part of the condition (C) holds and consequently, (C) holds for the function j . Since the function uR belongs to H 1 (W), we have kqk uR k2 6 clk¡ 1 ;
k = 1; 2;: :: :
Therefore, the function y satis es condition (C). 5. APPLICATIONS TO FINITE ELEMENT CONVERGENCE ESTIMATES Let W be a convex polygonal domain in R2 , with boundary ¶ W and no right angles. Let w be the measure of the largest corner and let s0 = minf1; p =w ¡ 1g and let u 2 H01 (W) be the variational solution of (4.1) with f 2 L2 (W). We let Vh to be
New regularity estimates
91
a nite dimensional approximation subspace of H01 (W) and consider the discrete problem: Find uh 2 Vh such that Z
W
Ñuh ¢ Ñv dx =
Z
W
f v dx
(5.1)
8 v 2 Vh :
Further, let us assume that
and
ku ¡ uh kH 1 (W) 6 chkukH 2 (W)
8 u 2 V 2 (W) = H 2 (W) \ H01 (W)
(5.2)
ku ¡ uh kH 1 (W) 6 ch2 kukH 3 (W)
8 u 2 V 3 (W) = H 3 (W) \ H01 (W):
(5.3)
By interpolation with p = 2 and 0 < s < 1, from (5.3) and (5.2), we obtain that ku ¡ uh kH 1 (W) 6 ch1+s kuk[V 3 (W);V 2 (W)]1¡
s;2
(5.4)
for all u 2 [V 3 (W);V 2 (W)]1¡ s;2 , where c is a constant independent of h. Interpolating with p = ¥ and s = 1 ¡ s0 we have ku ¡ uh kH 1 (W) 6 ch1+s0 kuk[V 3 (W);V 2 (W)]1¡
s0 ;¥
(5.5)
for all u 2 [V 3 (W);V 2 (W)]1¡ s0 ;¥ where again c is a constant independent of h. We thus have the following result. Theorem 5.1. Let u; uh be the variational solutions of problem (4.1) and (5.1), respectively. Then, there exists a constant c independent of h such that ku ¡ uh kH 1 6 ch1+sk f kH s
8 f 2 H s ; 0 < s < s0
(5.6)
8 f 2 Bs10 :
(5.7)
ku ¡ uh kH1 6 ch1+s0 k f kBs0 1
Furthermore, for s0 < s 6 1 there exists a constant c independent of h and s such that ku ¡ uh kH 1 (W) 6 c(s ¡ s0 )¡ 1=2 h1+s0 k f kHs (W) 8 f 2 H s(W) (5.8) and for h 6 e¡
1=(2(1¡ s0 ))
,
ku ¡ uh kH 1 (W) 6 ch1+s0 (log 1=h)1=2 k f kH s0 (W)
8 f 2 H s0 (W):
(5.9)
Proof. The inequalities (5.6) and (5.7) follows from (5.4) and (5.5), respectively, as a direct consequence of Theorem 4.1. The estimate (5.8) follows from
C. Bacuta, J. H. Bramble, and J. Xu
92
(5.7) and (3.6) with e = s ¡ s0 . The inequality (5.9) is obtained from (5.8) as follows. Let f 2 H s0 (W) and write f = f ¡ Qh f + Qh f . Next, we denote by uh 2 H01 (W) the solution of Z Z Ñuh ¢ Ñv dx = Qh f v dx 8 v 2 H01 (W): W
W
Since u ¡ uh is the solution of (4.1) with f ¡ Qh f instead of f we have ku ¡ uh kH 1 (W) 6 ck f ¡ Qh f kH ¡
1 (W)
and from standard approximation properties we get k f ¡ Qh f kH ¡
1 (W)
6 ch1+s0 k f kH s0 (W) :
Combining the two inequalities we have ku ¡ uh kH 1 (W) 6 ch1+s0 k f kH s0 (W) :
(5.10)
Next, using the estimate (5.8), a standard inverse inequality and the stability of the L2 projection in H s0 (W), we get kuh ¡ uh kH 1 (W) 6 c(s ¡ s0 )¡ 1+s0
6 ch
1=2 1+s0
h
kQh f kH ¡
(s ¡ s0 )
¡ 1=2 s0 ¡ s
6 ch1+s0 (s ¡ s0 )¡
h
1=2 s0 ¡ s
h
1+s (W)
kQh f kHs0 (W) k f kH s0 (W)
where c is a constant independent of of h and s. The minimum of the function s ! (s ¡ s0 )¡ 1=2 hs0 ¡ s on the interval (s0 ;1] is (2e log 1=h)1=2 and is attained for s = s0 + (2 log 1=h)¡ 1 , with h 6 e¡ 1=(2(1¡ s0 )) . Thus, kuh ¡ uh kH 1 (W) 6 ch1+s0 (log 1=h)1=2 k f kH s0 (W) :
(5.11)
Finally, (5.9) follows from (5.10) and (5.11).
¤
REFERENCES 1. C. Bacuta, J. H. Bramble, and J. Pasciak, New interpolation results and applications to nite element methods for elliptic boundary value problems. Numer. Lin. Algebra Appl. (to appear). 2. C. Bacuta, J. H. Bramble, and J. Xu, Regularity estimates for elliptic boundary value problems in Besov spaces. Math. Comp. (to appear). 3. C. Bennett and R. Sharpley, Interpolation of Operators. Academic Press, New York, 1988. 4. J. Bergh and J. Lostr ¨ om, ¨ Interpolation Spaces. An Introduction. Springer-Verlag, New York, 1976. 5. D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge, 1997.
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6. J. H. Bramble, Interpolation between Sobolev spaces in Lipschitz domains with an application to multigrid theory. Math. Comp. (1995) 64, 1359 – 1365. 7. J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation.SIAM J. Numer. Anal. (1970) 7, 113 – 124. 8. J. H. Bramble, J. Pasciak and P. S. Vassilevski, Computational scales of Sobolev norms with applications to preconditioning. Math. Comp. (2000) 69, 463 – 480. 9. J. H. Bramble, J. Pasciak, and J. Xu, Parallel multilevel preconditioners.Math. Comp. (1990) 55, 1 – 22. 10. J. H. Bramble and X. Zhang, The analysis of multigrid methods. In: Handbook for Numerical Analysis (Eds. P. Ciarlet and J.-L. Lions). North Holland, Amsterdam, 2000, Vol. VII, pp.173 – 415. 11. S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. SpringerVerlag, New York, 1994. 12. P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. 13. M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, Vol. 1341. Springer-Verlag, Berlin, 1988. 14. M. Dauge, S. Nicaise, M. Bourland, and M. S. Lubuma, Coef cients des singulariti´es pour des problemes ` aux limities elliptiques sur un domaine a` points coniques. I. Resultats ´ gen ´ eraux ´ pour le probl`eme de Dirichlet. RAIRO Model. ´ Math. Anal. Numer. ´ (1990) 24, 27 – 52. 15. M. Dauge, S. Nicaise, M. Bourland, and M. S. Lubuma, Coef cients des singulariti´es pour des problemes ` aux limities elliptiques sur un domaine a` points coniques. II. Quelques ope´erateurs particuliers. RAIRO Model. ´ Math. Anal. Numer. ´ (1990) 24, 343 – 367. 16. M. Dauge, S. Nicaise, M. Bourland, and M. S. Lubuma, Coef cients of the singularities for elliptic boundary value problems on domains with conical points. III. Finite Element Methods on Polygonal Domains. SIAM J. Numer. Anal. (1992) 29, No. 1, 136 – 155. 17. M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems. Rheinischen Friedrich-Wilhelms-Universit¨at, Bonn, 1981. 18. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985. 19. P. Grisvard, Singularities in Boundary Value Problems. Masson, Paris, 1992. 20. P. Grisvard, Caracterisation de quelques espaces d’interpolation. Arc. Rat. Mech. Anal. (1967) 25, 40 – 63. 21. R. B. Kellogg, Interpolation between subspaces of a Hilbert space. Tech. note No. BN-719. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, 1971. 22. V. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. (1967) 16, 227 – 313. 23. V. A. Kozlov, V. G. Mazya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Mathematical Surveys and Monographs, 1997, Vol. 52. 24. J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, New York, 1972. 25. J.-L. Lions and P. Peetre, Sur une classe d’espaces d’interpolation. Institut des Hautes Etudes Scienti que. Publ. Math. (1964) 19, 5 – 68. 26. S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth
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Boundaries. Expositions in Mathematics, de Gruyter, New York, 1994, Vol. 13. 27. J. Necas, Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967. 28. P. Oswald, Multilevel Finite Element Approximation. B. G. Teubner, Stuttgart, 1994. 29. L. Wahlbin, On the sharpness of certain local estimates for H01 projections into nite element spaces: In uence of a reentrant corner. Math. Comp. (1984) 42, 1 – 8. 30. J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Review (1992) 34, 581 – 613.
Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains C. BACUTA ¤ , J. H. BRAMBLE †, and J. Xu¤ Received 10 December, 2002
Communicated by R. Lazarov
Abstract — We consider the model Dirichlet problem for Poisson’s equation on a plane polygonal convex domain W with data f in a space smoother than L2 . The regularity and the critical case of the problem depend on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates for the critical case. As a consequence, sharp error estimates for the corresponding discrete problem are proved. Some classical shift estimates are also proved using the powerful tools of interpolation theory and mutilevel approximation theory. The results can be extended to a large class of elliptic boundary value problems. Keywords: interpolation spaces, nite element method, multilevel decomposition, shift theorems, subspace interpolation
1. INTRODUCTION Regularity estimates of the solutions of elliptic boundary value problems in terms of Sobolev norms of fractional order are known as shift theorems or shift estimates. The shift estimates for the Laplace operator with Dirichlet boundary conditions with non-smooth data on polygonal domains are well known (see, e.g, [2,21,23,27]). The classical regularity estimate for the case when W is a convex polygonal domain in R2 , with boundary ¶ W, is as follows: If u is the variational solution of ½ ¡ Du = f in W (1.1) u = 0 on ¶ W then, u 2 H 2 (W) and kukH 2 6 Ck f kH 0
8 f 2 H 0(W) = L2 (W):
(1.2)
Let w be the radian measure of the largest corner of W (w < p ), and let s0 = minf1; p =w ¡ 1g. If u is the variational solution of (1.1), then for 0 < s < s0 , it ¤ Dept. of Mathematics, The Pennsylvania State University University Park, PA 16802, USA † Dept. of Mathematics, Texas A & M University, College Station, TX 77843, USA The work of the second author was supported under NSF Grant No. DMS-9973328. The work of the rst and the third authors was supported under NSF Grant No.DMS-0209497.
C. Bacuta, J. H. Bramble, and J. Xu
76
is known (see e.g. [16]) that kukH 2+s 6 C(s)k f kH s
8 f 2 H s (W):
(1.3)
Here, H s(W) is the interpolation space between H1 (W) and L2 (W) s¡ distance from L2 (W). In this paper we will prove a stronger version of (1.3) and approach the problem in the critical value situation s = s0 . We prove that, for s0 < 1, the following estimate holds. kukB2+s0 (W) 6 ck f kBs0 (W) 8 f 2 Bs10 (W) (1.4) ¥
0 B2+s ¥ (W)
1
Bs10 (W)
where and are standard interpolation spaces de ned in Section 3. It is worth noting here that the space Bs10 (W) contains all the Hilbert spaces Hs with s > s0 . The estimate (1.4) leads to new nite element convergence estimates. The method presented in this paper can be extended to other boundary conditions such as Neumann or mixed Neumann–Dirichlet conditions. The technique involved in proving shift results is the real method of interpolation of Lions and Peetre ([3,24,25]), combined with multilevel approximation theory. The cases q = 2, q = 1 and q = ¥, where q is the second index of interpolation, are of special importance for our problem. The following type of interpolation problem is essential for our approach. If X and Y are Sobolev spaces of integer order and XK is a subspace of codimension one of X , then how can one characterize the interpolation spaces between XK and Y for q = 2 and q = ¥? The problem was studied in [20,21] and [1] for q = 2 and particular spaces X and XK . This paper gives an interpolation result of this type for the case q = ¥ . The remaining part of the paper is structured as follows. The interpolation results presented in Section 2 give a formulas for the norms on the intermediate subspaces [XK ;Y ]s;2 and [XK ;Y ]s;¥ when XK is of codimension one. In Section 3 the main result concerning the codimension-one subspace interpolation problem is presented. Shift theorems for the Poisson equation on polygonal domains are considered in Section 4. In the last section, a straightforward application of the interpolation results is shown to lead to some new estimates for nite element approximations. 2. INTERPOLATION RESULTS In this section we give some de nitions and results concerning interpolation spaces via real method of interpolation of Lions and Peetre (see [3,4,24]). 2.1. Interpolation between Banach spaces Let X ;Y be Banach spaces satisfying for some positive constant c, ½ X is a dense subset of Y and kukY 6 ckukX 8 u 2 X where kukX and kukY are the norms on X and Y , respectively.
(2.1)
New regularity estimates
77
The interpolation spaces [X ;Y ]s;q , 1 6 q 6 ¥, and s 2 (0; 1) are de ned using the K function, where for u 2 Y and t > 0, K(t;u) := inf (ku0 k2X + t 2 ku ¡ u0 kY2 )1=2 : u0 2X
Then, for q < ¥, the space [X ;Y ]s;q consists of all u 2 Y such that Z ¥ 0
(t ¡ s K(t;u))q
dt 0
K(t;u)2 < ¥:
The norm on [X ;Y ]s;q is de ned by kuk2[X;Y ]s;q
:=
Z ¥ 0
(t ¡ s K(t; u))q
dt t
and the norm on [X ;Y ]s;¥ is de ned by kuk [X ;Y ]s;¥ := sup t ¡ s K(t;u): t>0
Remark 2.1. Since K(t; u) 6 tkukY for all u 2 Y , the interval (0;¥) in the above de nitions can be replaced by any subinterval (A; ¥). The new norms obtained are equivalent with the original norms. 2.2. Interpolation between Hilbert spaces An extended Hilbert interpolation theory can be found in [24]. For completeness and consistence of notation we present in this section the interpolation results used in this paper. Let X ;Y be separable Hilbert spaces with inner products (¢; ¢)X and (¢; ¢)Y , respectively, and satisfying (2.1). Let D(S) denote the subset of X consisting of all elements u such that (2.2) v ! (u; v)X ; v 2 X is continuous in the topology induced by Y . For any u in D(S) the anti-linear form (2.2) can be uniquely extended to a continuous anti-linear form on Y . Then by Riesz representation theorem, there exists an element Su in Y such that (u;v)X = (Su;v)Y 8 v 2 X : (2.3) In this way S is a well de ned operator with domain D(S) in Y . The next result illustrates the properties of S (see [24]).
C. Bacuta, J. H. Bramble, and J. Xu
78
Proposition 2.1. The domain D(S) of the operator S is dense in X and consequently D(S) is dense in Y . The operator S : D(S) » Y ! Y is a bijective, selfadjoint and positive de nite operator. The inverse operator S¡ 1 : Y ! D(S) » Y is a bounded symmetric positive de nite operator and (S ¡ 1 z; u)X = (z; u)Y
(2.4)
8 z 2 y; u 2 X :
The next lemma provides the relation between K(t;u) and the connecting operator S. A proof of the lemma is given in [2]. Lemma 2.1. For all u 2 Y and t > 0 , ¡ K(t;u)2 = t 2 (I + t 2 S¡ 1 )¡
1
¢ u; u Y :
For the special case q = 2 (X ; Y Hilbert spaces), due to the spectral or multilevel representation of the norm on [X ;Y ]s;2 , the de nition of the norm is slightly changed as follows: Z ¥ kuk2[X ;Y ]s;2 := c2s t ¡ (2s+1)K(t;u)2 dt 0
where
cs :=
µZ
0
¥
t 1¡ 2s dt t2 + 1
¶¡
1=2
=
r
2 sin(p s): p
With the new de nition for the norm of [X ;Y ]s;2 it is natural (see the multilevel representation case in Section 3) to de ne [X ;Y ]0;2 := X ;
[X ;Y ]1;2 := Y:
Remark 2.2. Lemma 2.1 yields other expressions for the norms on [X ;Y ]s;2 and [X ;Y ]s;¥ . Namely, 2
kuk and
[X;Y ]s;2
=
c2s
Z ¥ 0
t¡
2s+1
kuk2 [X;Y ]s;¥ = sup t 2(1¡ t>0
s)
¡
(I + t 2 S ¡ 1 )¡ 1 u;u
¡
(I + t 2 S¡
1 ¡ 1
¢
) u;u
Y
¢
Y
dt
(2.5)
:
(2.6)
2.3. Interpolation between subspaces of a Hilbert space Let k = spanfj g be a one-dimensional subspace of X and let Xk be the orthogonal complement of k in X in the (¢; ¢)X inner product. We are interested in determining the interpolation spaces of Xk and Y , where on Xk we consider again the (¢; ¢)X inner product. To apply the interpolation results from the previous section we need to check that the density part of the condition (2.1) is satis ed for the pair (Xk ;Y ).
New regularity estimates
79
For j 2 k , de ne the linear functional Lj : X ! C, by Lj u := (u; j )X ;
u 2 X:
The following result is an extension of Kellogg’s lemma [21]. The proof can be found in [1]. Lemma 2.2. Let k be a closed subspace of X and let Xk be the orthogonal complement of k in X in the (¢; ¢)X inner product. The space Xk is dense in Y if and only if the following condition is satis ed: ½ Lj is not bounded in the topology of Y (2.7) for all j 2 k ; j 6= 0: For the remaining part of this section we assume that Lj is not bounded in the topology of Y, so the condition (2.1) is satis ed for the pair (Xk ;Y ). We denote Xk by Xj . It follows from the previous section that the operator Sj : D(Sj ) » Y ! Y de ned by (u; v)X = (Sj u;v)Y 8 v 2 Xj (2.8) has the same properties as S. Consequently, the norms on the intermediate spaces [Xj ;Y ]s;2 and [Xj ;Y ]s;¥ are given by: 2
kuk and
[Xj ;Y ]s;2
=
c2s
Z ¥ 0
t¡
2s+1
kuk2 [Xj ;Y ]s;¥ := sup t 2(1¡
¡
s)
t>0
¢ (I + t 2 S¡j 1 )¡ 1 u; u Y dt
¡
(I + t 2 Sj¡ 1 )¡ 1 u;u
¢
Y
(2.9)
(2.10)
:
Our aim in this section is to determine suf cient conditions for j such that the norm [Xj ;Y ]s;q (for q = 2 and q = ¥) can be compared with more familiar intermediate norms which are independent of j . First, we note that the operators Sj and S are related by the following identity: Sj¡ 1 = (I ¡ Qj )S¡
1
(2.11)
where Qj : X ! k is the orthogonal projection onto k = spanfj g. The proof of (2.11) follows easily from the de nitions of the operators involved. Next, (2.11) leads to a new formula for the norms on [Xj ;Y ]s;¥ and [X ;Y ]s;2 . Theorem 2.1. For any u 2 Y we have, kuk2 [Xj ;Y ]s;2 = kuk2 [X ;Y ]s;2 + c2s
Z ¥ 0
t¡
2 2s+3 j(u; j )Y;t j
(j ; j )X;t
dt
(2.12)
C. Bacuta, J. H. Bramble, and J. Xu
80
and
Ã
kuk2[Xj ;Y ]s;¥ = sup t 2¡ t>0
2s
(u; u)Y;t + t 4¡
2 2s j(u; j )Y;t j (j ; j )X;t
!
:
(2.13)
dt
(2.14)
In particular for u 2 Xj we have kuk2 [Xj ;Y ]s;2 = kuk2 [X ;Y ]s;2 + c2s and
Ã
kuk2[Xj ;Y ]s;¥ = sup t 2¡ t>0
where
Z ¥ 0
t¡
(2s+1) j(u; j )X;t j
and
(j ; j )X;t
j(u; j )X;t j2 2s (u; u)Y;t + t ¡ 2s (j ; j )X;t
¡ ¢ (u; v)X;t := (I + t 2 S¡ 1 )¡ 1 u;v X ¡ ¢ (u; v)Y;t := (I + t 2 S¡ 1 )¡ 1 u; v Y
2
!
(2.15)
8 u;v 2 X
(2.16)
8 u;v 2 Y:
(2.17)
Proof. The rst two formulas follows immediately from Remark 2.2 and the following identity ¡
¢ j(u; j )Y;t j2 (I + t 2 S¡j 1 )¡ 1 u; u Y = (u; u)Y;t + t 2 : (j ; j )X;t
The proof of the identity is based on (2.11). A detailed proof of it can be found in [2]. Using (2.4) and a simple manipulation of the operator S we get t 2 (u; j )Y;t = (u; j )X ¡ (u; j )X;t which, for u 2 Xj leads to (2.14) and (2.15).
¤
Theorem 2.1 can be easily extended to the case when K is of nite dimension(see [1]). Such an extension would be needed, for example, to treat the case of mixed Neumann–Dirichlet conditions or the case of the biharmonic problem. 3. MULTILEVEL REPRESENTATION OF INTERPOLATION SPACES Let W be a domain in R2 with boundary ¶ W. Assume that M1 » M2 » ¢ ¢¢ » Mk » : :: is a sequence of nite dimensional subspaces of H1 (W) whose union is dense in H 1 (W), and assume that an equivalent norm on H1 (W) is given by à !1=2 kuk1 :=
¥
å lk k(Qk ¡
k=1
Qk¡ 1 )uk2
(3.1)
New regularity estimates
81
where Qk denotes the L2 (W) orthogonal projection onto Mk, k¢k = k¢kL2 (W) , Q 0 = 0 and lk = 4k¡ 1 . The sequence fMk g can be taken for example the standard sequence of piecewise linear functions associated with a sequence of nested meshes. Proofs for the multilevel representation of the norm on H1 , for speci c choices of the spaces Mk can be found in [1,10,28,30]. 3.1. Scales of multilevel norms On H 1 (W) we consider the norm given by (3.1) and de ne H¡ 1 (W) to be the dual of H 1 (W). The elements of L2 (W) can be viewed as continuous linear functionals on H 1 (W) and we have the natural continuous and dense embeddings H 1 (W) » L2 (W) » H ¡ 1 (W): The projection Qk , k = 1; 2;: :: , can be extended to be de ned on H¡ 1 (W) by 8 u 2 H ¡ 1 (W); v 2 Mk
(Qk u; v)L2 = (u; v)
where (¢; ¢) on the right hand side represents the duality between H¡ 1 (W) and H 1 (W). One can easily check that the induced inner product on He (W) is given by (u; v)e :=
¥
å lke (qk u; qk v)
8 u; v 2 H e (W); e = ¡ 1; 1
k=1
where qk = Qk ¡ Qk¡ 1 . Then the pair (H 1 (W);L2 (W)) satis es the condition (2.1) and the operator S associated with the pair is given by Su =
¥
å lk qk u
(3.2)
8 u 2 D(S):
k=1
Thus, for X = H 1(W) and Y = L2 (W), we have ¥
(u;v)Y;t =
lk
å lk + t 2 (qk u;qk v);
u; v 2 Y
k=1
and
¥
(u;v)X;t =
l2
å lk +k t 2 (qk u;qk v);
u; v 2 X :
k=1
For any s 2 (0; 1), q = 1 or q = ¥, let H s (W) := [H 1 (W); L2 (W)]1¡
s;2
;
H 2+s(W) := [H 3 (W); H 1 (W)]1¡
s;2
C. Bacuta, J. H. Bramble, and J. Xu
82
and Bsq (W) := [H 1 (W);L2 (W)]1¡
3 1 B2+s q (W) := [H (W); H (W)]1¡
s;q ;
s;q :
By using (2.5) and (2.6), one can easily check that kuk2H s (W) = and
¥
å lkskqk uk2
(3.3)
k=1
Ã
! l k kuk2Bs¥ (W) = sup t 2s å kqk uk2 : 2 l + t k t>0 k=1 ¥
(3.4)
One can verify using the above formula that B0¥ (W) = L2 (W) and B1¥ (W) = H 1 (W). In addition we have t 2s = ss (1 ¡ s)1¡ s lk¡ 1+s sup 2 t>0 lk + t which leads to kuk2Bs¥ (W) 6 ss (1 ¡ s)1¡ s kuk2Hs (W)
8 u 2 H s(W)
(3.5)
a well known embedding property. Remark 3.1. If we assume that the sequence of subspaces fMk g is chosen so that the following approximation and inverse inequalities hold with a constant c independent of j: (Ap)
ku ¡ Q j uk 6 c 2¡ j kukH 1
(Inv)
kuk 6 c 2 j kukH 1
8 u 2 H 1 (W)
8u 2 Mj
then, by a well known result of the approximation theory, an equivalent norm on [H 1 ; L2 ]1¡ s;q is given by kf2s j kq j ukg j>1 klq ;
0 < s < 1; 1 6 q 6 ¥:
In particular, an equivalent norm on Bs1 := [H 1 ;L2 ]1¡ ¥
å 2s j kq j uk;
s;1
is given by
0 < s < 1:
j=1
One can verify now that we have the following norm-inequality: kukBs1 6 ce ¡ with c independent of e .
1=2
kukH s+e
8 u 2 H s+e
(3.6)
New regularity estimates
83
3.2. Subspace interpolation results Next, we focus our attention on a speci c case of subspace interpolation associated with the pair (X ;Y ), where X = H 1(W) and Y = L2 (W). For a xed s0 in the interval (0;1), let J0 = 1 ¡ s0 and j 2 H ¡ 1 (W). By the Riesz representation theorem there exists a function j 2 H 1(W) such that (v; j ) = (v; j )1 =
¥
å lk (qk j ;v)
8 v 2 H 1 (W):
k=1
Since qi q j = 0 for i 6= j, we deduce that in fact qk j = lk¡ 1 qk j :
(3.7)
Next, we assume that the function j satis es the following condition: (C) There exist two positive constants c1 ;c2 such that c1 lks0 6 kqk j k2 6 c2 lks0 ;
k = 1; 2;: :: :
Note that the above condition is equivalent to c1 lk¡
J0
J0
6 kqk j k21 6 c2 lk¡
;
k = 1; 2;: :: :
Lemma 3.1. Let j 2 H ¡ 1 (W) satisfy (C) and let j be the corresponding H1 representation. Then the following conditions are also satis ed. (C.0) Hj1 is dense in [H 1 ; L2 ]1¡
s
for s < s0 . Here, Hj1 is the kernel of j .
(C.1) There exist two positive constants c1 ;c2 such that c 1t ¡
2J0
6 (j ; j )X;t =
¥
lk2 å lk + t 2 k qk j k2 6 c2t ¡ k=1
2J0
;
t > 1:
Proof. The constants involved in this proof might change at different occurrences. To prove (C.0), by Lemma 2.2, it is enough to show that the functional u ! (u; j )1 = (u; j );
u 2 H 1 (W)
(3.8)
is not continuous in the topology induced by Hs (W), (s < s0 ). Let fun g be the sequence in H 1 (W) de ned by un :=
n
å lk¡ s
k=1
0
qk j :
C. Bacuta, J. H. Bramble, and J. Xu
84
Then,
n
(un ; j )1 = (un ; j ) =
å lk¡ s kqk j k2 ! ¥
as n ! ¥:
0
k=1
On the other hand kun k2H s =
n
å lks¡
2s0
kqk j k2
k=1
is uniformly bounded. Therefore, the functional de ned in (3.8) is not continuous and (C.0) is proved. To estimate (j ; j )X;t we observe that kqk j k2 = lk¡ 1 kqk j k21 and that
lk1¡ J0 å lk + t 2 = t ¡ k=1 ¥
2J0
¥
å
k=1
(4k =t 2 )1¡ J0 : (4k =t 2 + 1)
Using the standard convergence criteria for sums via integrals, the last sum can be ¤ estimated below and above by constants which are independent of t. Next theorem is the main subspace interpolation result of the paper. Theorem 3.1. Let j 2 H ¡ 1 (W) satisfy (C). Then [Hj1 ;L2 ]1¡ and
s;2
= [H 1 ;L2 ]1¡
Bs10 := [H 1; L2 ]1¡
s;2
:= H s(W);
» [Hj1 ;L2 ]1¡
s0 ;1
s < s0
(3.9) (3.10)
s0 ;¥ :
Proof. Recall that L2 (W) = Y and H 1 (W) = X . In order to prove (3.9) it is enough (by the density property (C.0), (2.14) and Remark 2.1) to prove that I :=
Z ¥ 1
t¡
2 (2(1¡ s)+1)) j(u; j )X;t j
(j ; j )X;t
dt 6 ckuk2H s
8 u 2 Xj
s=2
for u 2 H s (W) denote u˜k := lk kqk uk and u˜ := fu˜k g. Then we have kukHs = kuk ˜ l2 : Here (¢; ¢) is simply the L2 (W) inner product. Then, we have ¡ (u; j )X;t = (I + t 2 S¡ 1 )¡
1
u; j
¢
X
¥
=
k=1
For u 2 Xj we have (u; j )X = 0. Then ¥
å (qk u; j )X = 0:
k=1
lk
å lk + t 2 (qk u; qk j )X :
(3.11)
New regularity estimates
Consequently,
85
¥
1 (q u;qk j ): 2 k l k=1 k + t
(u; j )X;t = ¡ t 2 å
Thus, using the Cauchy–Schwarz inequality and (C) we obtain the estimate
lks0 =2 kqk uk: 2 l k=1 k + t ¥
j(u; j )X;t j 6 c2 t 2 å
(3.12)
Now we are prepared to estimate the integral I. The constant c, to be used next, may have different values at different places in which it appears. Let s1 = s0 ¡ s. Then, by (C.1) and the estimate (3.12), we have !2 à Z ¥ ¥ l 1¡ s0 =2 kqk uk dt I6c t ¡ 3+2¡ 2s0 +4 å k 2 l 1 k=1 k + t à ! Z ¥ ¥ s0 =2 l l ( ) m n 6c kqm ukkqn uk dt t 3¡ 2s1 å 2 2 1 m;n=1 (lm + t )(ln + t ) ¥
=c
å
(lm ln )s0 =2 kqm uk kqn uk
m;n=1
Z ¥ 1
t 3¡ 2s1 dt: (lm + t 2 )(ln + t 2 )
Next, we use the formula Z ¥ 0
t 3¡ 2J 1 a1¡ J ¡ b1¡ dt = (a + t 2 )(b + t 2 ) a¡ b c2J
J
; 0 < J < 2; J 6= 1; a; b > 0: (3.13)
The integral can be calculated by elementary calculus methods. If a = b, then the right side of the above identity is replaced by 1¡c2J a¡ J . Thus, J
Z ¥ 1
t 3¡ 2s1 dt 6 (lm + t 2 )(ln + t 2 )
Z ¥ 0
l 1¡ s1 ¡ ln1¡ t 3¡ 2s1 dt = c¡s12 m 2 2 lm ¡ ln (lm + t )(ln + t )
Combining the above inequalities, we get I6c
¥
å
(lm ln )s1 =2
lm1¡
m;n=1
¡ ln1¡ lm ¡ ln s1
Let lmn = (lm ln )s1 =2
lm1¡
s1
lms=2 kqm uklns=2 kqn uk:
¡ ln1¡ lm ¡ ln s1
Then, the above estimate becomes I6c
¥
å
m;n=1
lmn u˜m u˜n :
s1
:
s1
:
C. Bacuta, J. H. Bramble, and J. Xu
86
An elementary calculation gives lmn =
n)(1¡ s1 ) ¡
2(m¡
2¡ n) ¡ 2¡
2(m¡
(m¡ n)(1¡ s1 ) (m¡ n)
jm¡ njs1
6 2¡
; m; n = 1;2; :: :
and by elementary estimates we obtain I 6 ckuk ˜ 2l2 = ck uk2H s which proves (3.9). Next, we prove (3.10). Let u 2 Bs10 . Then by (2.13) and Remark 2.1 we have that kuk 2[Xj ;Y ]1¡ Note that
s0 ;¥
Ã
6 c sup t 2s0 (u;u)Y;t + sup t 4¡ t>1
t>1
2 2J0 j(u; j )Y;t j (j ; j )X;t
!
:
sup t 2s0 (u; u)Y;t 6 sup t 2s0 (u; u)Y;t = kuk2Bs¥0 6 ckuk2Bs0 t>1
1
t>0
and, with the help of (C.1) we have sup t 4¡
2J0
t>1
j(u; j )Y;t j2 6 c sup t 4 j(u; j )Y;t j 2 : (j ; j )X;t t>1
(3.14)
For u 2 Y , we have (u; j )Y;t =
¥
¥
lk
1
å lk + t 2 (qk u;qk j ) = å lk + t 2 (qk u;qk j )
k=1
k=1
and by using condition (C), we obtain
lks0 =2 j(u; j )Y;t j 6 c2 å kqk uk: 2 l k=1 k + t ¥
The function
(3.15)
lks0 =2 kqk uk 2 l k=1 k + t ¥
t ! t 2 j(u; j )Y;t j = t 2 å
is an increasing function of t. As t ! ¥, the limit of the above function is exactly kukBs0 . Therefore, 1
and the proof is complete.
kuk2[Xj ;Y ]1¡
s0 ;¥
6 ckuk2[X ;Y ]1¡
s0 ;1
¤
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87
4. A SHIFT THEOREM FOR THE LAPLACE OPERATOR ON CONVEX POLYGONAL DOMAINS In this section we will prove a stronger version of the estimate (1.3) and (1.4). 4.1. Shift estimates Let W be a convex polygonal domain in R2 , with boundary ¶ W and no right angles. Let V k (W) := H k (W) \ H01(W), k = 2 and k = 3. It is well known that for f 2 L2 (W) the variational problem has a unique solution u 2 V2 (W) and (1.2) holds. If W is an acute triangular domain then one can prove that for f 2 H1 (W) the solution of (1.1) belongs to V 3 (W) and (1.3) holds for s = 1. Using (1.2) and interpolation we obtain kukH 2+s 6 c(s)kuk[V 3 ;V 2 ]1¡
s;2
6 C(s)k f kH s
8 f 2 H s(W); 0 6 s 6 1:
Thus, without restricting the generality of the problem, we can assume that there exist one corner of measure w with p =2 < w < p . In fact, by a partition of unity type argument and using the regularity results for domains with smooth boundary, we can reduce the problem to the case when W is a domain with only one corner of measure w with p =2 < w < p . We will call this the ‘w -corner’ and we will assume that the vertex of the w -corner coincides with the origin of polar system of coordinates. Let a = p =w and s0 = a ¡ 1. Given f 2 H 1 (W), we consider the Dirichlet problem (1.1). The variational formulation of (1.1) is : Find u 2 H01 (W) such that Z
W
Ñu ¢ Ñv dx =
Z
W
f v dx
8 v 2 H01 (W):
(4.1)
Let z 2 d(W) be a cut-off function, which depends only on the distance r to the w -corner, is identically equal to one in a neighborhood of the w -corner and is identically equal to zero close to the part of ¶ W which does not contain the sides of the w -corner. Let y = j + uR , j (r; J ) = z r ¡ a sin(aJ ) and uR 2 H01(W) be the variatonal solution of (1.1) with f = Dj . One can check without dif culty that y 2 H ¡ 1 (W). Let Hy1 be de ned as the kernel of y as linear functional on H1 . Then, (see e.g., Theorem 9.8 in [16]) for f 2 Hy1 the (variational) solution of (1.1) Belongs to V 3 (W) and kukH 3 (W) 6 ck f kH 1 (W)
8 u 2 Hy1 (W):
(4.2)
Thus, by interpolation, we have kuk[V 3 (W);V 2 (W)]1¡ for all f 2 [H 1 (W)y ;L2 (W)]1¡
s;q .
s;q
6 ck f k[H 1 (W)y ;L2 (W)]1¡
s;q
Then, we have the following theorem.
(4.3)
C. Bacuta, J. H. Bramble, and J. Xu
88
Theorem 4.1. Let W be a convex polygonal domain in R2 with no right angles and with w the measure of the largest angle and let s0 = minf1; p =w ¡ 1g. If u is the variational solution of (1.1) then, for 0 < s < s0 there exist positive constant c(s) and C(s) such that kukH2+s 6 c(s)kuk[V 3 ;V 2 ]1¡
s;2
6 C(s)k f kH s
8 f 2 H s (W)
(4.4)
and for 0 < s0 < 1, kukB2+s0 (W) 6 c(s0 )kuk[V 3 ;V 2 ]1¡ ¥
s0 ;¥
6 C(s0 )k f kBs0 (W) 1
8 f 2 Bs10 (W):
(4.5)
Proof. Use (4.3) and apply Theorem 3.1 with j = y . The proof that (C) is satis ed is given later. The lower part of (4.4) or (4.5) follows by comparing the K ¤ functions associated with the two intermediate spaces. 4.2. Proving condition (C) Let W be a polygonal convex domain with the only one vertex O of measure w with p =2 < w < p and the remaining vertices denoted by S1 ;S2 ; :: : Sn . Let W=
n [
ti
i=1
where, for i = 1; :: : ;n, ti is a triangular domain with vertices Si ; O; Si+1 and O is taken to be the origin of a Cartesian system of coordinates in the plane. For i = 1; :: : ; n+ 1; let Gi denote the segment [O;Si ]. We assume, without loss of generality, that S1 lies on the positive semi-axis (see Fig. 1, the case n = 2). Let t1 = ft1 ;: :: ; tn g be the initial triangulation of W. We de ne multilevel triangulations recursively. For k > 1, the triangulation tk is obtained from tk¡ 1 by splitting each triangle in tk¡ 1 into four triangles by connecting the midpoints of the edges. The space Mk is de ned to be the space of all functions which are piecewise linear with respect to tk , vanish on ¶ W and are continuous on W. Let Qk denote the L2 (W) orthogonal projection onto Mk . We verify that the function y satis es the condition (C). To begin with, we will prove that the function j satis es (C). First we will prove that there exist a function wh 2 Mk which is supported in a ball of radius H = 1=2k¡ 1 = 2h centered at origin and wh is orthogonal on the space Mk¡ 1 . Let j1 ; :: : ; j 7 be the nodal functions in Mk¡ 1 corresponding to the nodal points in tk¡ 1 marked by ‘¯’ in the gure. Next, we consider the eight nodal functions j1 ;: :: j8 corresponding to the nodes marked by ‘?’). We de ne wh to be a linear combination wh of j1 ;: :: ; j8 , with coef cients independent of h such that (wh ; j j ) = 0; j = 1;: :: ;7; Hence, wh is orthogonal on the space Mk¡
1
(wh ; j ) 6= 0:
and consequently qk wh = wh .
New regularity estimates
G3
89
G2 ¯
¯ ¤
¯
¤
¤ ¯ ¤
¤ O
¤ ¯
¤
¯
¤ ¯
G1
Figure 1. The w-corner of W.
Then, (qk j ;v) (j ;qk v) j(j ; qk wh )j j(j ;wh )j > = sup = : kwh k kwh k v2L2 (W) kvk v2L2 (W) kvk
kqk j k = sup Note further that
kwh k 6 ch with c independent of h and by making the change of variable x = hxˆ in the integral which de nes the inner product (j ; wh ) we get j(j ; wh )j > ch2¡
a
for another constant c. Combining the above estimates we conclude that the lower part of the condition (C) holds. 1=2 For the upper part of (C) we rst note that kqk j k0 = lk kqk j k¡ 1 . To estimate kqk j k¡ 1 we let h = hh to be a cut off function which depends only on r and satis es
hh (r) = 1 for r 6 h=2;
hh (r) = 0 for r > h
jhh0 (r)j 6 c=h; jhh00 (r)j 6 c=h2
8 h=2 6 r 6 h
for some positive constant c. Then, kqk j k¡
1
= sup v2H 1 (W)
(qk j;v) kvk1
6
sup v2H 1 (W)
(qk (hj);v) kvk1
= M1 + M2 :
+ sup v2H 1 (W)
(qk ((1¡ h)j);v) kvk1
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90
Using polar coordinates we have Z w µZ h hj ( ; qk v) = r¡ 0
a+1
0
¶ h qk v dr sin(aJ ) dJ :
Next, we integrate by parts with respect to the r variable (write r1¡ a as the derivative of r 2¡ a =(2 ¡ a )). Using the Cauchy–Schwarz inequality and the estimate for h0 we get ¡ ¢ j(hj ; qk v)j 6 c h¡ a+1 kqk vk + h¡ a+2 k(qk v)r k : The L2 and H 1 -stability of the L2 projection give kqk vk 6 chkvk1 ;
kqk vk1 6 ckvk1
with c independent of h (or k). Thus, j(hj ; qk v)j 6 ch2¡
a
M1 6 ch2¡
kvk1 ;
a
:
To estimate M2 , we observe rst that (1 ¡ hh )j 2 H 2 (W). Let Ph : H 2 (W) ! Mk be the interpolant associated with th = Tk . By applying standard approximation properties and (1.2) we obtain M2 = kqk (1 ¡ hh )j k¡
1
6 hkqk (1 ¡ hh )j k 6 hk(I ¡ Qk¡ 3
1 )(1 ¡ 3
h )j k
6 chk(I ¡ Ph )(1 ¡ hh )j k 6 ch k(1 ¡ hh )j kH 2 (W) 6 ch kD(1 ¡ hh )j )kL2 (W) : Using a simple computation in polar coordinates, and the estimates for the derivative of hh , we get kD(hh j )kL2 (W) 6 ch¡ 1¡ a : Combining the above inequalities, we have that M2 6 ch2¡
a
:
Hence, the upper part of the condition (C) holds and consequently, (C) holds for the function j . Since the function uR belongs to H 1 (W), we have kqk uR k2 6 clk¡ 1 ;
k = 1; 2;: :: :
Therefore, the function y satis es condition (C). 5. APPLICATIONS TO FINITE ELEMENT CONVERGENCE ESTIMATES Let W be a convex polygonal domain in R2 , with boundary ¶ W and no right angles. Let w be the measure of the largest corner and let s0 = minf1; p =w ¡ 1g and let u 2 H01 (W) be the variational solution of (4.1) with f 2 L2 (W). We let Vh to be
New regularity estimates
91
a nite dimensional approximation subspace of H01 (W) and consider the discrete problem: Find uh 2 Vh such that Z
W
Ñuh ¢ Ñv dx =
Z
W
f v dx
(5.1)
8 v 2 Vh :
Further, let us assume that
and
ku ¡ uh kH 1 (W) 6 chkukH 2 (W)
8 u 2 V 2 (W) = H 2 (W) \ H01 (W)
(5.2)
ku ¡ uh kH 1 (W) 6 ch2 kukH 3 (W)
8 u 2 V 3 (W) = H 3 (W) \ H01 (W):
(5.3)
By interpolation with p = 2 and 0 < s < 1, from (5.3) and (5.2), we obtain that ku ¡ uh kH 1 (W) 6 ch1+s kuk[V 3 (W);V 2 (W)]1¡
s;2
(5.4)
for all u 2 [V 3 (W);V 2 (W)]1¡ s;2 , where c is a constant independent of h. Interpolating with p = ¥ and s = 1 ¡ s0 we have ku ¡ uh kH 1 (W) 6 ch1+s0 kuk[V 3 (W);V 2 (W)]1¡
s0 ;¥
(5.5)
for all u 2 [V 3 (W);V 2 (W)]1¡ s0 ;¥ where again c is a constant independent of h. We thus have the following result. Theorem 5.1. Let u; uh be the variational solutions of problem (4.1) and (5.1), respectively. Then, there exists a constant c independent of h such that ku ¡ uh kH 1 6 ch1+sk f kH s
8 f 2 H s ; 0 < s < s0
(5.6)
8 f 2 Bs10 :
(5.7)
ku ¡ uh kH1 6 ch1+s0 k f kBs0 1
Furthermore, for s0 < s 6 1 there exists a constant c independent of h and s such that ku ¡ uh kH 1 (W) 6 c(s ¡ s0 )¡ 1=2 h1+s0 k f kHs (W) 8 f 2 H s(W) (5.8) and for h 6 e¡
1=(2(1¡ s0 ))
,
ku ¡ uh kH 1 (W) 6 ch1+s0 (log 1=h)1=2 k f kH s0 (W)
8 f 2 H s0 (W):
(5.9)
Proof. The inequalities (5.6) and (5.7) follows from (5.4) and (5.5), respectively, as a direct consequence of Theorem 4.1. The estimate (5.8) follows from
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92
(5.7) and (3.6) with e = s ¡ s0 . The inequality (5.9) is obtained from (5.8) as follows. Let f 2 H s0 (W) and write f = f ¡ Qh f + Qh f . Next, we denote by uh 2 H01 (W) the solution of Z Z Ñuh ¢ Ñv dx = Qh f v dx 8 v 2 H01 (W): W
W
Since u ¡ uh is the solution of (4.1) with f ¡ Qh f instead of f we have ku ¡ uh kH 1 (W) 6 ck f ¡ Qh f kH ¡
1 (W)
and from standard approximation properties we get k f ¡ Qh f kH ¡
1 (W)
6 ch1+s0 k f kH s0 (W) :
Combining the two inequalities we have ku ¡ uh kH 1 (W) 6 ch1+s0 k f kH s0 (W) :
(5.10)
Next, using the estimate (5.8), a standard inverse inequality and the stability of the L2 projection in H s0 (W), we get kuh ¡ uh kH 1 (W) 6 c(s ¡ s0 )¡ 1+s0
6 ch
1=2 1+s0
h
kQh f kH ¡
(s ¡ s0 )
¡ 1=2 s0 ¡ s
6 ch1+s0 (s ¡ s0 )¡
h
1=2 s0 ¡ s
h
1+s (W)
kQh f kHs0 (W) k f kH s0 (W)
where c is a constant independent of of h and s. The minimum of the function s ! (s ¡ s0 )¡ 1=2 hs0 ¡ s on the interval (s0 ;1] is (2e log 1=h)1=2 and is attained for s = s0 + (2 log 1=h)¡ 1 , with h 6 e¡ 1=(2(1¡ s0 )) . Thus, kuh ¡ uh kH 1 (W) 6 ch1+s0 (log 1=h)1=2 k f kH s0 (W) :
(5.11)
Finally, (5.9) follows from (5.10) and (5.11).
¤
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