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SIAM J. NUMER. ANAL. Vol. 43, No. 6, pp. 2320–2344

SOME NEW ERROR ESTIMATES OF A SEMIDISCRETE FINITE VOLUME ELEMENT METHOD FOR A PARABOLIC INTEGRO-DIFFERENTIAL EQUATION WITH NONSMOOTH INITIAL DATA∗ RAJEN K. SINHA† , RICHARD E. EWING‡ , AND RAYTCHO D. LAZAROV§ Abstract. A semidiscrete ﬁnite volume element (FVE) approximation to a parabolic integrodiﬀerential equation (PIDE) is analyzed in a two-dimensional convex polygonal domain. An optimalorder L2 -error estimate for smooth initial data and nearly the same optimal-order L2 -error estimate for nonsmooth initial data are obtained. More precisely, for homogeneous equations, an elementary energy technique and a duality argument are used to derive an error estimate of order O t−1 h2 ln h in the L2 -norm for positive time when the given initial function is only in L2 . Key words. parabolic equation, integro-diﬀerential equation, optimal-order error estimate, smooth and nonsmooth initial data AMS subject classiﬁcations. 65M12, 65M60, 65N40 DOI. 10.1137/040612099

1. Introduction. The aim of this paper is to analyze a semidiscrete ﬁnite volume element (FVE) method for solving initial-boundary value problems for an integrodiﬀerential equation of the form (1.1)

ut − ∇ · (A∇u) = −

t

∇ · (B∇u(s))ds + f (x, t) in Ω × J, 0

u = 0 on ∂Ω × J, u(·, 0) = u0 in Ω. Here, Ω ⊂ R2 is a bounded convex polygonal domain with boundary ∂Ω, J = (0, T ] with T < ∞, and ut = ∂u/∂t. Further, A = {ai,j (x)} is a symmetric and uniformly positive deﬁnite matrix of size 2 × 2 in Ω and B = {bi,j (x, t, s)} is a 2 × 2 matrix. The nonhomogeneous term f = f (x, t) and the coeﬃcients aij (x), bij (x; t, s) are assumed to be smooth for our purpose. For the sake of simplicity, we shall denote Au = −∇·(A∇u) and B(t, s)u(s) = −∇·(B∇u(s)). For references to studies regarding existence, uniqueness, and regularity of such problems, one may refer to [33]. Parabolic integro-diﬀerential equations (PIDEs) of the above type arise naturally in many applications, such as, for instance, heat conduction in materials with memory [27], nonlocal reactive ﬂows in porous media [10, 11], and non-Fickian ﬂow of ﬂuid in porous media [15]. One very important characteristic of these models is that they all ∗ Received

by the editors July 22, 2004; accepted for publication (in revised form) May 20, 2005; published electronically January 6, 2006. The work of the ﬁrst author was supported by the DST, Government of India under BOYSCAST fellowship. The work of the second and third authors was partially supported by the NSF-EIA grant 0218229. http://www.siam.org/journals/sinum/43-6/61209.html † Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781039, India ([email protected]). ‡ Institute for Scientiﬁc Computation, Texas A&M University, College Station, TX 77843-3404 ([email protected]). § Department of Mathematics, Texas A&M University, College Station, TX 77843-3404 (lazarov @math.tamu.edu). 2320

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express the conservation of a certain quantity (mass, momentum, heat, etc.) in any moment for any subdomain. This in many applications is the most desirable feature of the approximation method when it comes to numerical solution of the corresponding initial-boundary value problem. For references to studies of existence, uniqueness, and regularity of such problems, one may refer to [33]. To put our work into proper perspective, we ﬁrst give a brief account of the development of the ﬁnite element methods for such problems. Over the last decade, various numerical methods based on ﬁnite element approximations in space and special quadrature in time have been developed and studied for this type of problem [20, 24, 25, 29, 31, 32, 34]. The crucial tools used in the analysis are the Ritz and Ritz– Volterra projections which are instrumental in deriving optimal-order error estimates in various Sobolev norms [5, 6, 20]. In [31], the authors studied this type of problem for both smooth and nonsmooth initial data cases. In particular, for a homogeneous equation with nonsmooth initial data, an optimal-order L2 -error estimate is proved via a semigroup theoretic approach. Subsequently, using the energy method, the authors 2 2 of [25] derived convergence of order O ht for the L2 -norm and O ht log( h1 ) for the L∞ -norm for the homogeneous equation when the initial function is in H01 (Ω)∩H 2 (Ω). Recently, in [26], the analysis from [21] of the case B(t, s) = 0 was carried over to a time dependent PIDE. An optimal-order error estimate by energy techniques and a duality argument for the homogeneous equation with both smooth and nonsmooth initial data were carried over. In both [21] and [26], negative norm estimates are used in a crucial way in their analyses. In the absence of the memory term, i.e., when B(t, s) ≡ 0, the error estimates for ﬁnite element methods for both smooth and nonsmooth data cases are described in [2, 18, 28, 30] and the references cited therein. In recent years, the numerical methods for problem (1.1) by means of FVE discretizations were considered in [13] and [14]. The interest in such methods is due to certain conservation features of FVE methods that are desirable in many applications. In [13] and [14], the authors studied FVE approximation of such a problem in the framework of the standard Petrov–Galerkin formulation and obtained L2 -error estimate of the form (cf. [14, p. 305])

(1.2)

u(t) − uh (t) ≤ Ch2 (u0 3,p + u(t)3,p t + (u(s)3,p + ut (s)3,p )ds), p > 1, 0

where u and uh represent the solution of (1.1) and its FVE approximation, respectively. Note that the estimate (1.2) is optimal with respect to the approximation property, but its regularity requirement on the exact solution seems to be too high when compared with that for ﬁnite element methods. This is primarily due to the fact that the bounds in the L2 -norm of a new variant of the Ritz–Volterra projection (the so-called Petrov–Volterra projection introduced in [13, 14]) are not optimal with respect to the regularity of the solution. In this paper, we analyze the FVE method for the problem (1.1) and derive optimal-order L2 -error estimates for both smooth and nonsmooth initial data. For the homogeneous problem with smooth initial data, we are able to show an L2 -error estimate which is optimal with respect to the order of convergence as well as the regularity of the solution. This is exactly the result known for ﬁnite element methods (cf. [25]). More precisely, we prove an optimal-order L2 -error estimate for f = 0 and initial data u0 ∈ H 2 (Ω) ∩ H01 (Ω). This technique, quite new and promising, is based on improved estimates for a new variant of the Ritz–Volterra projection (see

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Theorems 3.1 and 3.2). The main concern of this paper is to prove an L2 -error estimate for the homogeneous equation (f = 0) with nonsmooth initial data. This is motivated by the fact that the solutions of a homogeneous linear parabolic equation have the so-called smoothing property. That is, the solution is smooth for positive time t, even when the initial data are not. In quantitative form, this may be expressed by the inequality u(t)α ≤ Ct−α/2 u0 ,

(1.3)

t ∈ J,

which is valid for any α ≥ 0. Here · α is a Sobolev norm. However, this is not the case with PIDEs as they have a limited smoothing property. This fact is proved in [31], where the inequality (1.3) is shown to be valid only for α ≤ 2. Since the smoothing property plays a signiﬁcant role in the error analysis in the semidiscrete solution, an attempt has been made in this paper to derive an L2 -error estimate for the FVE method when the initial data u0 is only in L2 (Ω). More importantly, our analysis uses only energy techniques and a duality argument. The proposed techniques have several attractive features. Unlike the analyses of [21] and [26], we do not require error estimates in negative-indexed Sobolev norms while dealing with L2 -error estimates with nonsmooth initial data. Thus, these results hold for convex polygonal domains with corners, unlike [21] and [26]. Since the FVE method is thought of as a perturbation of the Galerkin ﬁnite element method, the proposed technique can easily be adopted to the ﬁnite element method as well. However, to the best of our knowledge the error estimates for nonsmooth initial data using the FVE method were not established earlier. The previous work on the theoretical framework and the basic tools for the analysis of the FVE methods for elliptic and parabolic problems are described in [3, 4, 9, 7, 8, 12, 16, 17, 19, 22, 23] and references therein. The outline of this paper is as follows. In section 2, we introduce some notation, formulate FVE approximations for piecewise linear ﬁnite element spaces deﬁned on a triangulation, and recall some basic estimates from the literature. Further, the Ritz–Volterra projection is introduced and related estimates are obtained in section 3. Section 4 is devoted to the error estimates for smooth initial data. Finally, error estimates with nonsmooth initial data are carried out in section 5. Throughout this paper C denotes a generic positive constant which does not depend on the mesh parameter h but may depend on T . 2. Notation and preliminaries. Let H01 (Ω) = φ ∈ H 1 (Ω) | φ = 0 on ∂Ω . Further, let A(·, ·) and B(t, s; ·, ·) be the bilinear forms on H01 (Ω) × H01 (Ω) given by (2.1) A(u, v) = A(x)∇u · ∇vdx; B(t, s; u(s), v) = B(x, t, s)∇u(s) · ∇vdx. Ω

Ω

For the purpose of FVE approximations we now consider the following weak formulation: Find u : J¯ → H01 (Ω) such that (2.2)

t

B(t, s; u(s), v)ds + (f, v) ∀v ∈ H01 (Ω), t ∈ J,

(ut , v) + A(u, v) = 0

with u(0) = u0 . Here and below, (·, ·) and · denote the L2 inner product and the induced norm on L2 (Ω). Further, we shall use the standard notation for Sobolev spaces W m,p (Ω)

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with 1 ≤ p ≤ ∞. The norm on W m,p (Ω) is deﬁned by |Dα u|p dx , 1 ≤ p < ∞, um,p,Ω = um,p = Ω |α|≤m

with the standard modiﬁcation for p = ∞. When p = 2, we write W m,2 (Ω) by H m (Ω) and denote the norm by · m . Further, H −1 (Ω) denotes the space of all bounded linear functionals on H01 (Ω). For a functional f ∈ H −1 (Ω), its action on a function u ∈ H01 (Ω) is denoted by (f, u), which represents the duality pairing between H −1 (Ω) and H01 (Ω). To simplify notation, we use (·, ·) to denote both the L2 (Ω) inner product and the duality pairing between H −1 (Ω) and H01 (Ω). 2.1. A priori estimates. In the following lemmas, we state some a priori bounds for the solution u satisfying (1.1) under appropriate regularity assumptions on the initial function u0 . For a proof, one may refer to [25, 26, 21]. Lemma 2.1. Let u satisfy (1.1). If u0 ∈ L2 (Ω) and f ∈ L2 (Ω), then

t t 2 2 u(t)2 + u(s)1 ds ≤ C u0 2 + f (s) ds . 0

0

and f ∈ L (Ω), we have Moreover, when u0 ∈

t t u(t)21 + {us (s)2 + u(s)22 }ds ≤ C u0 21 + f (s)2 ds . H01 (Ω)

2

0

0

Lemma 2.2. Let u satisfy (1.1). If u0 ∈ H 2 (Ω) ∩ H01 (Ω) and f ∈ L2 (Ω), then

t t 2 2 2 2 ut (t) + us (s)1 ds ≤ C ut (0) + f (s) ds . 0

0

Lemma 2.3. Let u satisfy (1.1) with f = 0, and let 0 ≤ i, j, k ≤ 2. If 0 ≤ k + 2j − i ≤ 2, then j 2 i ∂ u 2 t j (t) ≤ Cu0 k+2j−i . ∂t k Further, if 0 ≤ k + 2j − i − 1 ≤ 2, then 2 t j i ∂ u 2 s j (s) ds ≤ Cu0 k+2j−i−1 . ∂s 0 k 2.2. FVE approximation. Let Th be a quasi-uniform family of triangulations ¯ = ∪K∈T K, where K is a closed triangle element. Let Nh be the set of Ω such that Ω h of all nodes or vertices of Th , i.e., ¯ Nh = {p : p is a vertex of element K ∈ Th and p ∈ Ω}. Further, we denote Nh0 = Nh ∩ Ω. For a vertex xi ∈ Nh , let Π(i) be the index set of those vertices that, along with xi , are in some element of Th . For a given triangulation Th , we now introduce a dual mesh Th∗ as follows: In each element K ∈ Th with vertices xi , xj , and xk , select a point q ∈ K, select a point

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RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

V i

Vi x ij

xi

x ij xj

xj

xi

γ ij

γ ij

Fig. 2.1. Control volumes with barycenter as internal point and interface γij of Vi and Vj .

xij on the edge connecting xi and xj , and connect q with xij by straight lines γij,K . Then for a vertex xi we let Vi be the polygon whose edges are γij,K in which xi is a vertex of the element K. We call this Vi a control volume centered at xi . Further, we ¯ Thus, the dual mesh T ∗ is then deﬁned as the collection of note that ∪xi ∈Nh Vi = Ω. h these control volumes. A control volume centered at a vertex xi is given in Figure 2.1. We call the control volume mesh Th∗ regular or quasi-uniform if there exists a positive constant C > 0 such that C −1 h2 ≤ meas(Vi ) ≤ Ch2 ∀ Vi ∈ Th∗ , where h is the maximum diameter of all elements K ∈ Th . There are various ways to introduce a regular dual mesh Th∗ depending on the choices of the point q in an element K ∈ Th and the points xij on its edges. In this paper, we choose q to be the barycenter of an element K ∈ Th , and the points xij are chosen to be the midpoints of the edges of K. Obviously, if Th is regular, i.e., there is a constant C such that Ch2K ≤ meas(K) ≤ h2K , where hK = diam(K) for all elements K ∈ Th , then the dual mesh Th∗ is also regular. For the purpose of FVE approximation, let Sh be the standard linear ﬁnite element space deﬁned on the triangulation Th , Sh = {v ∈ C(Ω) : v|K is linear ∀ K ∈ Th and v|∂Ω = 0}, and its dual volume element space Sh∗ , Sh∗ = {v ∈ L2 (Ω) : v|V is constant ∀ V ∈ Th∗ and v|∂Ω = 0}. Obviously, Sh = span{φi (x) : xi ∈ Nh0 } and Sh∗ = span{ψi (x) : xi ∈ Nh0 }, where φi are the standard nodal basis functions associated with the node xi , and ψi are the characteristic functions of the volume Vi . Let Ih : C(Ω) → Sh and Ih∗ : C(Ω) → Sh∗ be the usual interpolation operators, i.e., Ih u(x) = ui φi (x) and Ih∗ u(x) = ui ψi (x), xi ∈Nh

where ui = u(xi ).

xi ∈Nh

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The FVE approximation is then deﬁned to be the function uh : J¯ → Sh such that t (2.3) (uht , Ih∗ χ) + A(uh , Ih∗ χ) = B(t, s; uh (s), Ih∗ χ)ds + (f, Ih∗ χ) ∀χ ∈ Sh . 0

Here uh (0) = P˜h u0 , where P˜h u0 is the L2 -projection of u0 onto Sh deﬁned by (P˜h u0 , Ih∗ χ) = (u0 , Ih∗ χ) ∀χ ∈ Sh ,

(2.4)

the bilinear forms A(·, ·) and B(t, s; ·, ·) in (2.3) are deﬁned by vi A(x)∇u · ndSx , A(u, v) = − ∂Vi

xi ∈Nh

B(t, s; u, v) = −

B(x, t, s)∇u · ndSx

vi

xi ∈Nh

∂Vi

for (u, v) ∈ ((H01 ∩ H 2 ) ∪ Sh ) × Sh∗ , and n is the outer-normal vector of the involved integration domain. Note that when (u, v) ∈ H01 (Ω)×H01 (Ω), the bilinear forms A(·, ·) and B(t, s; ·, ·) are given by (2.1). In order to describe features of the bilinear forms deﬁned in (2.2) and (2.3), we use some discrete norms on Sh and Sh∗ , meas(Vi )((uhi − uhj )/d2ij , |uh |20,h = (uh , uh )0,h , |uh |21,h = xi ∈Nh xj ∈Π(i)

uh 21,h

=

|uh |20,h

+

|uh |21,h ,

|uh |2 = (uh , Ih∗ uh ),

where (uh , vh )0,h = xi ∈Nh meas(Vi )uhi vhi = (Ih∗ uh , Ih∗ vh ) and dij = d(xi , xj ) is the Euclidean distance between xi and xj . The discrete norms | · |0,h and · 1,h are equivalent to the usual norms · and · 1 , respectively, on Sh . Some properties of the bilinear forms are stated below without proof. For a proof, see, e.g., [1, 12, 14]. Lemma 2.4. There exist positive constants C1 and C2 such that for all vh ∈ Sh , we have C1 |vh |0,h ≤ vh ≤ C2 |vh |0,h , C1 |vh | ≤ vh ≤ C2 |vh |, C1 vh 1,h ≤ vh 1 ≤ C2 vh 1,h . Lemma 2.5. There exist positive constants C and c such that for all φh , ψh ∈ Sh , we have |A(φh , Ih∗ ψh )| ≤ Cφh 1 ψh 1 , |B(t, s; φh , ψh )| ≤ Cφh 1 ψh 1 , and A(φh , Ih∗ φh ) ≥ cφh 21 . Lemma 2.6. If the matrix A(x) is constant over each element K ∈ Th , then we have A(uh , χ) = A(uh , Ih∗ χ) ∀uh , χ ∈ Sh .

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Following the arguments of Lemma 2.3 on the discrete level, it is easy to derive the following stability estimates for the FVE solution uh satisfying (2.3). Lemma 2.7. Let uh satisfy (2.3) with f = 0. Then we have

t

uh (t)2 +

uh (s)21 ds ≤ Cuh (0)2 , 0

t

suhs (s) + tuh (t)21 ds ≤ Cuh (0)2 , 0 t 2 2 s2 uhs (s)21 ds ≤ Cuh (0)2 . t uht (t) + 2

0

The following lemma gives the key feature of the bilinear forms in the FVE method. For a proof, see [12] or [7]. Lemma 2.8. Let φ ∈ H01 (Ω). Then we have ∗ (A∇φ · n)(χ − Ih∗ χ)dS A(φ, χ) = A(φ, Ih χ) + −

K∈Th

∂K

(∇ · A∇φ)(χ − Ih∗ χ)dx ∀χ ∈ Sh .

K

K∈Th

The above identity holds true when A(·, ·) is replaced by B(t, s; ·, ·). Remark 2.9. We note that the above identity is proved in [12, 7] for φ, χ ∈ Sh . In fact, identities in Lemma 2.8 holds true even if φ ∈ H01 (Ω). 3. Ritz–Volterra projection and related estimates. Following Lin et al. [20], we deﬁne the Ritz–Volterra projection Wh u of a function u(x, t) deﬁned on Ω × J¯ in the context of the FVE method and obtain bounds for the error in H 1 and L2 norms. The Ritz–Volterra projection Wh : L∞ (H01 ∩ H 2 ) → L∞ (Sh ) is deﬁned by (3.1)

A((u − Wh u)(t), Ih∗ χ) =

t

¯ B(t, s; (u − Wh u)(s), Ih∗ χ)ds ∀ χ ∈ Sh , t ∈ J.

0

Below, we shall prove a lemma which is frequently used in our subsequent analysis. Lemma 3.1. For any function φ ∈ H r (Ω)(r = 0, 1), we have (3.2)

|(φ, χ − Ih∗ χ)| ≤ Ch1+r φr χ1 ∀ χ ∈ Sh .

Further, for φ ∈ H01 (Ω), we have (3.3)

|A(φ, χ − Ih∗ χ)| ≤ Chφ1 χ1 ∀ χ ∈ Sh .

The second inequality also holds true when A(·, ·) is replaced by B(t, s; ·, ·). Proof. We borrow the proof of (3.2) from [8]. To show (3.3), we have [7] (∇ · A∇φ)(χ − Ih∗ χ)dx A(φ, χ − Ih∗ χ) = − (3.4)

+

K∈Th

K

K∈Th

∂K

((A − A¯K )∇φ · n)(χ − Ih∗ χ)dS.

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Here, A¯K is a function designed in a piecewise manner such that for any edge E of a triangle K ∈ Th and x ∈ E, A¯K (x) = A(xc ), where xc is the midpoint of E. Applying the Cauchy–Schwarz inequality and using the fact that χ−Ih∗ χ ≤ Chχ1 and |A(x) − A¯K | ≤ hA1,∞ , we obtain |A(φ, χ − Ih∗ χ)| ≤ Chφ1 χ1 , and this completes the proof. Set ρ = u − Wh u. We now establish the H 1 -error estimate for ρ and its temporal derivative. Theorem 3.1. Let Wh u be deﬁned by (3.1). Then we have

t

ρ(t)1 ≤ Ch u(t)2 + u(s)2 ds , 0

t u(s)2 ds . ρt (t)1 ≤ Ch u(t)2 + ut (t)2 + 0

Proof. With φh = Ih u − Wh u, we have cρ21 ≤ A(ρ, ρ) = A(ρ, u − Ih u) + A(ρ, Ih u − Wh u) t B(t, s; ρ(s), Ih∗ φh )ds. = A(ρ, u − Ih u) + A(ρ, φh − Ih∗ φh ) + 0

An application of (3.3) yields

cρ21

t

≤ Ch(u2 + u1 )ρ1 + C

ρ1 ds (ρ1 + hu2 ),

0

where for the last term on the right we have used the fact that φh 1 ≤ C(ρ1 + hu2 ). Kicking back ρ1 , we get

t ρ21 ≤ C h2 u22 + ρ21 ds . 0

Now applying Gronwall’s lemma, we obtain the ﬁrst inequality. To estimate ρt 1 , we diﬀerentiate (3.1) with respect to time t to get (3.5)

A(ρt , Ih∗ χ) = B(t, t, ρ(t), Ih∗ χ) +

t

Bt (t, s; ρ(s), Ih∗ χ)ds.

0

As before, with φh = Ih ut − Wh ut we obtain cρt 21 ≤ A(ρt , ρt ) = A(ρt , ut − Ih ut ) + A(ρt , φh − Ih∗ φh ) + B(t, t, ρ(t), Ih∗ φh ) t + Bt (t, s; ρ(s), Ih∗ φh )ds. 0

Now apply (3.3), the estimate of ρ1 , and the standard kickback argument to obtain the second inequality. Next, we derive L2 estimates for ρ = u − Wh u and its temporal derivative in the following theorem.

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Theorem 3.2. Let Wh u be deﬁned by (3.1). Then we have

t ρ(t) ≤ Ch2 u(t)2 + u(s)2 ds , 0

t 2 ρt (t) ≤ Ch u(t)2 + ut (t)2 + u(s)2 ds . 0

Proof. The proof will proceed by the duality argument. For t ∈ (0, T ) let ψ(t) ∈ H 2 (Ω) ∩ H01 (Ω) be the solution of (3.6)

Aψ = ρ in Ω, ψ = 0 on ∂Ω,

satisfying the following regularity estimate (recall that Ω is convex): ψ2 ≤ Cρ.

(3.7)

Multiplying (3.6) by ρ and then integrating over Ω, we obtain ρ2 = A(ρ, ψ − Ih ψ) + A(ρ, Ih ψ − Ih∗ (Ih ψ)) t t B(t, s; ρ(s), Ih∗ (Ih ψ) − Ih ψ)ds + B(t, s; ρ(s), Ih ψ − ψ)ds + 0 0 t B(t, s; ρ(s), ψ)ds = I1 + I2 + I3 + I4 + I5 . + 0

In view of Theorem 3.1, I1 and I4 are bounded as

t u2 ds ψ2 . |I1 | + |I4 | ≤ Ch2 u2 + 0

For I2 and I3 , an application of Lemma 3.3 and Theorem 3.1 yields

t t ρ1 ds ψ1 ≤ Ch2 u2 + u2 ds ψ1 . |I2 | + |I3 | ≤ Ch ρ1 + 0

0

Finally, I5 is estimated as t

t ∗ ρds ψ2 , |I5 | ≤ (ρ(s), B (t, s)ψ) ds ≤ C 0

0

where B ∗ (t, s) is the adjoint of B(t, s). Now putting these estimates together and with an aid of (3.7) we obtain

t t ρ = Ch2 u2 + u2 ds + C ρds. 0

0

Finally, an application of Gronwall’s lemma yields the ﬁrst estimate. To estimate ρt , we again use the duality argument, (3.5), and the estimate of ρ to complete the proof. Remark 3.2. (i) The estimates in Theorem 3.2 are optimal with respect to the order of convergence as well as the regularity requirement on the solution. This

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improves upon the earlier result of [13] and [14] by requiring less regularity on the solution. (ii) In the absence of an integral term (when B(t, s) = 0), as a consequence of Theorems 3.1 and 3.2, error estimates associated with the Petrov–Ritz projection Rh : H01 → Sh deﬁned by ¯ A(Rh u − u, Ih∗ χ) = 0 ∀ χ ∈ Sh , t ∈ J, can easily be obtained. Thus, we immediately have (3.8)

Rh u − u + hRh u − u1 ≤ Chj uj , u ∈ H01 (Ω) ∩ H j (Ω), j = 1, 2.

Below, we shall prove a lemma which is crucial for the error estimate in the case of nonsmooth initial t data to be discussed in section 5. Deﬁne ρˆ(t) = 0 ρ(τ )dτ . Then, integrating by parts we rewrite (3.1) as t A(ˆ ρt (t), Ih∗ χ) = B(t, s; ρˆs (s), Ih∗ χ)ds 0 t Bs (t, s; ρˆ(s), Ih∗ χ)ds. = B(t, t, ρˆ, Ih∗ χ) − 0

Integrate from 0 to t to obtain t t (3.9) A(ˆ B(s, s, ρˆ(s), Ih∗ χ)ds − ρ(t), Ih∗ χ) = 0

0

s

Bs (s, τ ; ρˆ(τ ), Ih∗ χ)dτ ds.

0

Lemma 3.2. Let u be the solution of the initial value problem (1.1) with f = 0 t and ρˆ(t) = 0 (u − Wh u)(s)ds. Then we have ˆ ρ + hˆ ρ1 ≤ Ch2 u0 . Proof. With φh = Ih u ˆ − Wh u ˆ, we have cˆ ρ21 ≤ A(ˆ ρ, ρˆ) = A(ˆ ρ, u ˆ − Ih u ˆ) + A(ˆ ρ, Ih u ˆ − Wh u ˆ) t ˆ) + A(ˆ ρ, φh − Ih∗ φh ) + B(s, s; ρˆ(s), Ih∗ φh )ds ≤ A(ρ, u ˆ − Ih u 0 t s Bs (s, τ ; ρˆ(τ ), Ih∗ φh )dτ ds, − 0

where u ˆ(t) = we obtain

t 0

0

u(s)ds. Then proceeding as in the estimate of ρ1 in Theorem 3.1

u2 + ˆ ρ1 ≤ Ch ˆ

(3.10)

t

ˆ u2 ds .

0

Now it remains to estimate ˆ u2 . From (1.1) with f = 0, we have t t B(t, s)ˆ us (s)ds = − ut + B(t, t)ˆ u(t) − Bs (t, s)ˆ u(s)ds. Au = − ut + 0

0

Integrating from 0 to t and then using elliptic regularity and Lemma 2.1, we obtain t t ˆ u2 ≤ u0 + u(t) + C ˆ u2 ds ≤ Cu0 + C ˆ u2 ds. 0

0

2330

RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

Now an application of Gronwall’s lemma yields ˆ u2 ≤ Cu0 .

(3.11)

Combine (3.10) and (3.11) to obtain ˆ ρ1 . Next, using (3.9), the proof technique of ρ in Theorem 3.2, and (3.11), the estimate of ˆ ρ can be easily obtained. This completes the rest of the proof. 4. Error estimates for problems with smooth initial data. In this section, we estimate the error of the semidiscrete FVE method for problems with smooth initial data. In particular, an optimal-order L2 -error estimate is obtained when u0 ∈ H01 (Ω) ∩ H 2 (Ω). As usual we write the error e(t) = u(t) − uh (t) as a sum of two terms e(t) = (u − Wh u) + (Wh u − uh ) = ρ + θ. The estimate of ρ is already established, so it is enough to estimate θ. Using (2.3), an equation of the form (2.3) with uh replaced by u, and (3.1), it is easy to verify that θ satisﬁes an error equation of the form t ∗ ∗ (θt , Ih χ) + A(θ, Ih χ) = B(t, s; θ(s), Ih∗ χ)ds − (ρt , Ih∗ χ) ∀χ ∈ Sh . (4.1) 0

Analogously, integrating (2.3) from 0 to t and then using the resulting equation with uh replaced by u, (3.9), and uh (0) = P˜h u0 , we obtain an error equation in θˆ as t ∗ ∗ ˆ ˆ ˆ (θt , Ih χ) + A(θ, Ih χ) = B(s, s; θ(s), Ih∗ χ)ds 0 t s ˆ ), I ∗ χ)dτ ds − (ρ, I ∗ χ), χ ∈ Sh , (4.2) Bτ (s, τ ; θ(τ − h h 0

0

t

ˆ = θ(s)ds. Below, we shall prove a sequence of lemmas that will lead us where θ(t) 0 to the desired result. Lemma 4.1. There is a positive constant C such that t t 2 2 2 ˆ ˆ + θ(s) ds ≤ C ρ(s) ds. θ(t) 1 0

0

Proof. Choose χ = θˆ in (4.2) to have t 1 d ˆ ∗ˆ ˆ ˆ I ∗ θ) ˆ = ˆ B(s, s; θ(s), Ih∗ θ(t))ds (θ, Ih θ) + A(θ, h 2 dt 0 t s ˆ ), I ∗ θ)dτ ˆ ds − (ρ, I ∗ θ). ˆ Bτ (s, τ ; θ(τ − h

0

h

0

Integrating from 0 to t and using the standard kickback argument yield t t t s 2 2 2 ˆ ˆ )2 dτ ds. ˆ θ(s)1 ds ≤ C ρ(s) ds + C θ(τ θ(t) + 1 0

0

0

0

Finally, apply Gronwall’s lemma to complete the rest of the proof. Lemma 4.2. There is a positive constant C such that t t 2 2 2 ˆ θ(s) ds + θ(t) ≤ C ρ(s) ds. 1 0

0

SEMIDISCRETE FINITE VOLUME ELEMENT METHOD

2331

Proof. Take χ = θ in (4.2) and integrate from 0 to t to have t t s 1 ˆ ∗ˆ ˆ ), I ∗ θ(s))dτ ds Ih θ) = (θ, Ih∗ θ)ds + A(θ, B(τ, τ ; θ(τ h 2 0 0 0 t s τ ˆ ), I ∗ θ(s))dτ dτ ds − (ρ, I ∗ θ) Bτ (τ, τ ; θ(τ − h h 0

0

0

= I1 + I2 + I3 . For I1 , we note that t t ˆ ), I ∗ θˆs (s))dsdτ I1 = B(τ, τ ; θ(τ h 0 τ t t ∗ˆ ˆ ˆ ), I ∗ θ(τ ˆ ))dτ. B(τ, τ ; θ(τ ), Ih θ(t))dτ ds − B(τ, τ ; θ(τ = h 0

0

Similarly, we rewrite the term I2 . Now use the standard kickback argument to obtain t t t ˆ 2≤C ˆ 2 ds + C θ(s)2 ds + θ θ ρ2 ds. 1 1 0

0

0

Finally, an application of Lemma 4.1 completes the rest of the proof. Lemma 4.3. There is a positive constant C such that t t 2 2 2 2 tθ(t) + sθ(s)1 ds ≤ C {ρ(s) + s2 ρs (s) }ds. 0

0

Proof. Take χ = tθ in (4.1) and integrate by parts to have 1 d 1 ˆ I ∗ θ(t)) {t(θ, Ih∗ θ)} + tA(θ, Ih∗ θ) = (θ, Ih∗ θ) + tB(t, t; θ(t), h 2 dt 2 t ˆ − tBs (t, s; θ(s), Ih∗ θ(t))ds − t(ρt , Ih∗ θ). 0

Integrating from 0 to t and applying the standard kickback argument, we obtain t t t s 2 2 2 ˆ ˆ )2 dτ ds tθ(t) + c sθ(s)1 ds ≤ C θ(s) ds + C θ(τ 1 1 0 0 0 0 t 2 2 {θ + s2 ρs }ds. +C 0

Then use Lemmas 4.1 and 4.2 to complete the proof. The main result of this section is given in the following theorem. Theorem 4.1. Let u and uh , respectively, satisfy (1.1) and (2.3) with f = 0. Then for u0 ∈ H 2 (Ω) ∩ H01 (Ω) and uh (0) = P˜h u0 , we have e(t) ≤ Ch2 u0 2 . Proof. By the triangle inequality, we write t1/2 e(t) ≤ t1/2 ρ(t) + t1/2 θ(t).

2332

RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

By Theorem 3.2 and a priori estimates in Lemma 2.3, the ﬁrst term on the right is bounded by t1/2 ρ(t) ≤ Ch2 t1/2 u0 2 . For the second term, we use Lemma 4.3, Theorem 3.2, and a priori estimates in Lemma 2.3 to have

t 1/2 2 2 1/2 2 t θ ≤ C {ρ(s) + s ρs (s) }ds ≤ Ch2 t1/2 u0 2 . 0

Altogether these estimates yield the desired result and this completes the proof. Remark 4.4. Note that the result presented in Theorem 4.1 is optimal with respect to the approximation property as well as the regularity of the solution. Similar result for ﬁnite element methods is established in [31, 25, 26]. 5. Error estimates for nonsmooth initial data. In this section we establish one of the main results of the paper, namely, an error estimate for problems with nonsmooth initial data. More precisely, an almost optimal-order L2 -error estimate is obtained when u0 ∈ L2 (Ω). The following lemma is useful in our subsequent analysis. Lemma 5.1. For all χ1 , χ2 ∈ Sh , we have u − χ1 1 )χ2 1 , A(χ1 , χ2 − Ih∗ χ2 )| ≤ Ch2 (χ1 1 + h−1/2 ˆ t where u ˆ(t) = 0 u(s)ds. The above estimate also holds true when A(·, ·) is replaced by B(t, s; ·, ·). Proof. From (3.4), we have ∗ A(χ1 , χ2 − Ih χ2 ) = − (∇ · A∇χ1 )(χ2 − Ih∗ χ2 )dx +

K∈Th

K

K∈Th

∂K

((A − A¯K )∇χ1 · n)(χ2 − Ih∗ χ2 )dS = I1 + I2 .

Since the dual mesh is formed by the barycenters, we have (χ − Ih∗ χ)dx = 0 ∀χ ∈ Th , K

and hence, we apply the Cauchy–Schwarz inequality to have |I1 | = ∇ · A∇χ1 − (∇ · A∇χ1 )K (χ2 − Ih∗ χ2 ) dx ≤ Ch2 χ1 1 χ2 1 , K K∈Th

1 where (∇ · A∇χ1 )K = area(K) ∇ · A∇χ1 dx. Since ∇ˆ u · n is continuous across any K edge E ∈ Th , we may rewrite I2 as I2 = (A − A¯K )∇(ˆ u − χ1 ) · n (χ2 − Ih∗ χ2 )dS, K∈Th

∂K

and hence using the fact that |A(x)−A¯K | ≤ hA1,∞ , the Cauchy–Schwarz inequality, and trace results, we obtain |I2 | ≤ Ch ∇(ˆ u − χ1 )L2 (∂K) χ2 − Ih∗ χ2 L2 (∂K) ≤ Ch3/2 ˆ u − χ1 1 χ2 1 . K∈Th

SEMIDISCRETE FINITE VOLUME ELEMENT METHOD

2333

Combining these estimates we complete the proof. Below, we shall prove several lemmas which will be used to derive error estimates for problems with nonsmooth initial data. Lemma 5.2. Let u and uh be the solution of (1.1) and (2.3), respectively. Then for u0 = 0, we have

t

u(s) − 0

2 uh (s)1 ds

f (0) +

≤ Ch

2

t

2

f ds .

2

0

Proof. Set χ = Rh e in the error equation (et , Ih∗ χ) + A(e, Ih∗ χ) =

(5.1)

t

B(t, s; e(s), Ih∗ χ)ds

0

to get 1 d 2 |e| + A(e, e) = (et , u − Ih∗ (Rh u)) + A(e, u − Ih∗ (Rh u)) 2 dt t − B(t, s; e(s), Ih∗ (Rh u) − Ih∗ uh )ds + (et , Ih∗ uh − uh ) 0

+ A(e, Ih∗ uh − uh ) = I1 + I2 + I3 + I4 + I5 . By (3.8), (3.2), and (3.3), we have |I1 | + |I2 | ≤ |(et , u − Rh u)| + |(et , Rh u − Ih∗ (Rh u))| + |A(e, u − Rh u)| + |A(e, Rh u − Ih∗ (Rh u))| ≤ Ch2 (et u2 + et 1 u1 ) + Che1 (u2 + u1 ). Again, in view of (3.2) and (3.3), I4 and I5 can be estimated as |I4 | + |I5 | ≤ C(h2 et 1 + he1 )uh 1 . To estimate I3 , we ﬁrst rewrite it as I3 =

t

B(t, s; e(s), Ih∗ (Rh u)

t

− Rh u)ds + B(t, s; e(s), Rh u − u)ds 0 0 t t B(t, s; e(s), e)ds + B(t, s; e(s), uh − Ih∗ uh )ds. + 0

0

Apply (3.3) and (3.8) to obtain

|I3 | ≤ Ch

t

t e(s)1 ds (u1 + u2 + uh 1 ) + C e(s)1 ds e1 .

0

0

Combining these estimates now leads to 1 d 2 e + A(e, e) ≤ Ch2 et u2 + Ch2 et 1 (u1 + uh 1 ) + Che1 u1 2 dt

t

t +h e1 ds (u1 + u2 + uh 1 ) + C e(s)1 ds e1 . 0

0

2334

RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

Integrate from 0 to t, use the fact that e(0) = 0, and then apply the standard kickback argument to obtain t t t s 2 2 2 2 2 2 e(s)1 ds ≤ Ch (uh 1 + u2 + et + et 1 )ds + C e(τ )21 dτ ds. 0

0

0

0

The desired estimate now easily follows from Lemmas 2.1 and 2.2, its discrete analogue, and Gronwall’s t lemma. Deﬁne eˆ(t) = 0 e(s)ds. Then, as a consequence of Lemmas 4.2 and 3.2 and a priori estimates we have the following lemma. Lemma 5.3. Assume that u0 ∈ L2 (Ω) and f = 0. Then there is a positive constant C independent of h such that ˆ e(t)1 ≤ Chu0 . In order to obtain optimal L2 -error estimate for problems with nonsmooth data, it is convenient to prove an estimate of ˆ e. For this purpose, we now consider the following backward problems. For ﬁxed time t > 0 and given any f¯ ∈ L2 (Ω), let v(s) ∈ H 2 (Ω) ∩ H01 (Ω) be the solution of the backward problem t vs − Av = − (5.2) B ∗ (τ, s)v(τ )dτ + f¯, s ≤ t, s

with v(t) = g, where B ∗ (τ, s) is the adjoint of B(τ, s). The associated weak solution is then deﬁned to be the function v : [0, t) → H01 (Ω) such that t (5.3) (φ, vs ) − A(φ, v) = − B(τ, s; φ, v(τ ))dτ + (φ, f¯) ∀φ ∈ H01 (Ω), s ≤ t, s

with v(t) = g. Analogous to (2.3), the FVE approximation is then deﬁned to be the function vh : [0, t) → Sh such that t ∗ ∗ (Ih χ, vhs ) − A(Ih χ, vh ) = − (5.4) B(τ, s; Ih∗ χ, vh (τ ))dτ + (Ih∗ χ, f¯) s

for all χ ∈ Sh , s ≤ t, with vh (t) = gh , where gh is a suitable approximation of g in Sh to be deﬁned later. Remark 5.4. With a simple change of variables in the proofs of Lemmas 2.1–2.3 and using the backward Gronwall lemma, it is easy to obtain a priori bounds for the backward solutions v and vh . Lemma 5.4. Assume that u0 ∈ L2 (Ω) and f = 0. Then there is a generic constant C such that ˆ e(t) ≤ Ch2 u0

(5.5)

∀t > 0.

Proof. Let w(s) ∈ H 2 (Ω) ∩ H01 (Ω) be the solution of the backward problem (5.2) with f¯ = eˆ and g = 0. Then, with a change of variables in the proofs of Lemmas 2.1– 2.3 and its discrete analogue, Lemma 5.2 and using the backward Gronwall lemma, it is an easy exercise to check that the solution w(s) and its FVE solution wh (s), which may be stated in a manner similar to (5.3)–(5.4), satisfy the following estimate: t t (5.6) {ws − whs 2 + ws − whs 21 + h−2 w − wh 21 + w22 }ds ≤ C ˆ e2 ds. 0

0

SEMIDISCRETE FINITE VOLUME ELEMENT METHOD

2335

We take an L2 inner product of (5.2) with e and use (5.1) to obtain 1 d d ˆ e(s)2 = (e, Ih∗ wh ) + (e, ws − whs ) − (e, whs − Ih∗ whs ) − A(e, w − wh ) 2 ds ds t ∗ −A(e, wh − Ih wh ) + B(τ, s; e(s), (w − wh )(τ ))dτ s t t B(τ, s; e(s), (wh − Ih∗ wh )(τ ))dτ + B(τ, s; e(s), Ih∗ wh (τ ))dτ + s s s − B(s, τ ; e(τ ), Ih∗ wh (s))dτ. 0

With η = u − Ih∗ (Rh u) and ζ = Ih∗ uh − uh , we rewrite the above equation as 1 d d ˆ e(s)2 = (e, Ih∗ wh ) + (η, ws − whs ) − A(η, w − wh ) 2 ds ds t + B(τ, s; η(s), (w − wh )(τ ))dτ + (e, whs − Ih∗ whs ) − A(e, wh − Ih∗ wh ) s t + B(τ, s; e(s), wh (τ ) − Ih∗ wh (τ ))dτ + (ζ, ws − whs ) − A(ζ, w − wh ) s t + B(τ, s; ζ(s), (w − wh )(τ ))dτ. s

Multiply both sides by s and integrate from 0 to t to have 1 1 tˆ e(t)2 = 2 2

t

ˆ e(s) ds + 2

0

t

(e, Ih∗ wh )ds

0

t

t

0

t

s(η, ws − whs )ds

+ 0

t

sA(η, w − wh )ds +

−

sB(τ, s; η, (w − wh ))(τ )dτ ds 0

t

s t

s(e, whs − Ih∗ whs )ds − sA(e, wh − Ih∗ wh )ds 0 t t s sB(s, τ ; e(τ ), Ih∗ wh (s) − wh (s))dτ ds + s(ζ, ws − whs )ds − 0 0 0 t t t sA(ζ, w − wh )ds + sB(τ, s; ζ(s), (w − wh )(τ ))dτ ds −

+

0

1 = 2

0

0 t

ˆ e(s)2 ds + 0

10

s

Ii .

i=1

Since eˆ(0) = 0 = Ih∗ w(t), we obtain using (5.6) t t ∗ |I1 | = − (ˆ e, Ih whs )ds ≤ C ˆ ewhs ds 0 0 1/2 t 1/2

t t 2 2 ˆ e ds whs ds ≤C ˆ e(s)2 ds. ≤ 0

0

0

2336

RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

For I2 , an application of (3.2), (3.8), (5.6), and a priori estimates yield t t ∗ |I2 | = s(u − Rh u, ws − whs )ds + s(Rh u − Ih (Rh u), ws − whs )ds 0 0 t t (s2 u22 + su21 )ds + C ≤ Ch4 ws − whs 21 ds 0 0 t ˆ e2 ds. ≤ Ch4 tu0 2 + C 0

Similarly, for I3 and I4 , using (3.8), (3.3), and (5.6), we obtain t t su − Rh u1 w − wh 1 ds + h |I3 | + |I4 | ≤ C sRh u − Ih∗ (Rh u)1 w − wh 1 ds 0 0 t t 4 2 2 2 2 −2 (s u2 + s u1 )ds + Ch w − wh 21 ds ≤ Ch 0 0 t 4 2 2 ˆ e ds. ≤ Ch tu0 + C 0

Apply (3.2), (5.6), and a priori estimates to have t t t se21 ds + C whs 21 ds ≤ Ch4 tu0 2 + C ˆ e2 ds. |I5 | = Ch4 0

0

0

For I7 , with a change of variables and integration by parts, we note that t t t s sB(τ, s; e(s), (wh − Ih∗ wh )(τ ))dτ ds = τ B(s, τ ; eˆτ (τ ), (wh − Ih∗ wh )(s))dτ ds 0 s 0 0 t = sB(s, s; eˆ(s), (wh − Ih∗ wh )(s))ds 0 t s τ Bτ (s, τ ; eˆ(τ ), (wh − Ih∗ wh )(s))dτ ds − 0 0 t s B(s, τ ; eˆ(τ ), (wh − Ih∗ wh )(s))dτ ds. − 0

0

Similarly, we rewrite the term I6 as t t I6 = sA(ˆ e, whs − Ih∗ whs )ds + A(ˆ e, wh − Ih∗ wh )ds, 0

0

where we have used the fact that wh (t) = 0 = eˆ(0). Thus, applying (3.3), Lemma 5.3, and (5.6), I6 and I7 are bounded by t t |I6 | + |I7 | ≤ Cth4 u0 2 + C (wh 21 + whs 21 )ds ≤ Cth4 u0 2 + C ˆ e2 ds. 0

0

Finally, using (3.2), (3.3), and (5.6), we obtain t t |I8 | + |I9 | + |I10 | ≤ Ch4 suh 21 ds + C (ws − whs 21 + h−2 w − wh 21 )ds 0 0 t ≤ Ch4 tu0 2 + C ˆ e2 ds. 0

2337

SEMIDISCRETE FINITE VOLUME ELEMENT METHOD

Altogether this now leads to tˆ e(t) ≤ Ch4 tu0 2 + C

(5.7)

It now remains to estimate respect to x to have ˆ e(s)2 =

(5.8)

t 0

t

ˆ e(s)2 ds. 0

ˆ e ds. Multiply (5.2) by eˆ and integrate by parts with 2

d e, ws − whs ) − A(ˆ e, w − wh ) (ˆ e, Ih∗ wh ) + (ˆ ds t + B(τ, s; eˆ(s), w(τ ) − wh (τ ))dτ + (ˆ e, whs − Ih∗ whs ) − A(ˆ e, wh − Ih∗ wh ) s s s τ − B(τ, τ ; eˆ(τ ), Ih∗ wh (s))dτ + Bτ (τ, τ ; eˆ(τ ), Ih∗ wh (s))dτ dτ 0 0 0 t B(τ, s; eˆ(s), wh (τ ))dτ. + s

Here, we have used the relation t t ∗ ∗ ∗ (e, Ih χ) + A(ˆ e, Ih χ) = B(s, s; eˆ(s), Ih χ)ds − 0

0

s

Bτ (s, τ ; eˆ(τ ), Ih∗ χ)dτ ds,

0

which is obtained by integrating (5.1) from 0 to t and using (2.4). Now integrate (5.8) from 0 to t and use the fact that eˆ(0) = 0 = Ih∗ wh (t) to have t t t ˆ e(s)2 ds = (ˆ η , ws − whs )ds − A(ˆ η , w − wh )ds 0 0 0 t t B(τ, s; ηˆ(s), (w − wh )(τ ))dτ ds − 0 s t s − B(τ, τ ; eˆ(τ ), Ih∗ wh (s))dτ ds 0 0 t s τ Bτ (τ, τ ; eˆ(τ ), Ih∗ wh (s))dτ dτ ds + 0 0 0 t t (ˆ e(s), whs − Ih∗ whs )ds − A(ˆ e, wh − Ih∗ wh )ds + 0 0 t t t ˆ ws − whs )ds B(τ, s; eˆ(s), wh (τ ))dτ ds + (ζ, + 0 s 0 t t t ˆ w − wh )ds − ˆ A(ζ, B(τ, s; ζ(s), (w − wh )(τ ))dτ ds − 0

=

11

0

s

Ji ,

i=1

where ηˆ = u ˆ − Ih∗ (Rh u ˆ) and ζˆ = Ih∗ u ˆh − u ˆh . Let us estimate each term separately. For J1 , use of (3.8), (3.11), (5.1), and (5.6) yields t |J1 | ≤ {|(ˆ u − Rh u ˆ, ws − whs )| + |(Rh u ˆ − Ih∗ (Rh u ˆ), ws − whs )|}ds 0 t t t ˆ u22 ds + ws − whs 2 ds ≤ C( )th4 u0 2 + C ˆ e2 ds. ≤ C( )h4 0

0

0

2338

RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

Similarly, |J2 | + |J3 | ≤ C( )th4 u0 2 + h−2

t

w − wh 21 ds ≤ C( )th4 u0 2 + C 0

ˆ e2 ds. 0

For J4 , we note that t s B(τ, τ ; eˆ(τ ), Ih∗ wh (s) − wh (s))dτ ds J4 = − 0 0 t s t B(τ, τ ; eˆ(τ ), wh (s) − w(s))dτ ds − − 0

t

0

0

s

B(τ, τ ; eˆ(τ ), w(s))dτ ds,

0

and hence, using (3.3), Lemma 5.3, and (5.6), we obtain t s {hˆ e(τ )1 (wh (s)1 + h−1 w(s) − wh (s)1 ) + ˆ e(τ )w(s)2 }dτ ds |J4 | ≤ C 0 0 t s ˆ e(τ )2 dτ ds. ≤ Cth4 u0 2 + C 0

0

Similarly,

t

s

|J5 | ≤ Cth4 u0 2 + C

ˆ e(τ )2 dτ ds. 0

0

Using (3.1)–(3.3), Lemma 5.3, and (5.6), we obtain t t |J6 | + |J7 | ≤ C( )h2 ˆ e21 ds + {whs 21 + wh 21 }ds 0 0 t 4 2 ˆ e2 ds. ≤ Cth u0 + C 0

By changing the order of integration, rewrite the term J8 as t s t s J8 = B(τ, s; eˆ(s), (wh − w)(τ ))dsdτ + B(τ, s; eˆ(s), w(τ ))dsdτ. 0

0

0

In view of Lemma 5.3 and (5.6), we obtain

t

0

s

|J8 | ≤ Cth4 u0 2 + C

ˆ e(τ )2 dτ ds. 0

0

Finally, using (3.2), (3.3), and (5.6), we have t |J9 | + |J10 | + |J11 | ≤ C( )th4 u0 2 + (ws − whs 21 + h−2 w − wh 21 )ds 0 t ˆ e(s)2 ds. ≤ C( )th4 u0 2 + C 0

Putting these estimates together and choosing appropriately, we arrive at t s t 2 4 2 ˆ e(s) ds ≤ Cth u0 + C ˆ e(τ )2 dτ ds. 0

0

0

An application of Gronwall’s lemma yields t ˆ e2 ds ≤ Cth4 u0 2 , 0

and this combined with (5.7) completes the rest of the proof.

SEMIDISCRETE FINITE VOLUME ELEMENT METHOD

2339

Remark 5.5. Deﬁning the error e¯ = v − vh associated twith the backward problem (5.3) and its FVE approximation (5.4), set ˜e¯(s) = − s e¯(τ )dτ , s ≤ t. Then, for g ∈ L2 (Ω) and f¯ = 0, analogous to Lemmas 5.3 and 5.4, it is easy to show that ˜e¯j ≤ Ch2−j g, j = 0, 1.

(5.9)

We conclude this section by showing our main result in the following theorem. Theorem 5.1. Let u and uh be solutions of (1.1) and (1.4), respectively, with f = 0. Assume that u0 ∈ L2 (Ω) and the matrix A is constant over each element K ∈ Th . Then there is a generic positive constant C independent of h such that e(t) ≤ Ct−1 h2 ln hu0 ,

t ∈ J.

Proof. Using (5.3) and (5.4) with f¯ = 0 and Lemma 2.6, we ﬁrst note that d 2 s [(u, v) − (Ih∗ uh , vh )] ds

s = 2s {(u, v) − (Ih∗ uh , vh )} + s2 B(s, τ ; u(τ ), v(s))dτ 0 s t s2 B(τ, s; u(s), v(τ ))dτ − s2 B(s, τ ; uh (τ ), Ih∗ vh (s) − vh (s))dτ − s 0 t s s2 B(τ, s; Ih∗ uh (s) − uh (s), vh (τ ))dτ − + s2 B(s, τ ; uh (τ ), vh (s))dτ s 0 t s2 B(τ, s; uh (s), vh (τ ))dτ − s2 (uhs , vh − Ih∗ vh ) − s2 (Ih∗ uhs − uhs , vh ). + s

Integrate the above equation from 0 to t. Then, with gh = Lh g, where Lh : L2 (Ω) → Sh deﬁned by (Lh g, Ih∗ χ) = (g, χ), χ ∈ Sh , we have 2

t

s {(u(s), v(s)) − (uh (s), vh (s))} ds t s s2 B(s, τ ; uh (τ ), Ih∗ vh (s) − vh (s))dτ − 0 0 t t t s2 B(τ, s; Ih∗ uh (s) − uh (s), vh (τ ))dτ − + s2 (uhs , vh − Ih∗ vh )ds 0 s 0 t s2 (uhs − Ih∗ uhs , vh )ds = 2I1 + I2 + I3 + I4 + I5 . −

t (e(t), g) = 2

0

(5.10)

0

For the term I2 , with u ˆh (t) = t I2 = − 0

t 0 s

uh (s)ds, we integrate by parts to have

s2 B(s, τ ; u ˆhτ (τ ), (Ih∗ vh − vh )(s))dτ ds

0 t

s2 B(s, s; u ˆh (s), (Ih∗ vh − vh )(s))ds t s s2 Bτ (s, τ ; u ˆh (τ ), (Ih∗ vh − vh )(s))dτ ds +

=−

0

0

0

= I21 + I22 .

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RAJEN SINHA, RICHARD EWING, AND RAYTCHO LAZAROV

For I21 , apply Lemma 5.1 with χ1 = u ˆh and χ2 = vh , Lemma 5.3, and a priori estimates to obtain t s2 (ˆ uh 1 + h−1/2 ˆ e1 )vh (s)1 ds ≤ Cth2 u0 g. |I21 | ≤ Ch2 0

The term

I22

is treated in a similar manner and hence |I2 | ≤ Cth2 u0 g.

t Similarly, deﬁning v˜h (s) = − s vh (τ )dτ, s ≤ t, we rewrite the term I3 as t t I3 = s2 B(τ, s; Ih∗ uh (s) − uh (s), v˜h,τ (τ ))dτ ds 0 s t s2 B(s, s; Ih∗ uh (s) − uh (s), v˜h (s))ds =− 0 t t s2 Bτ (τ, s; Ih∗ uh (s) − uh (s), v˜h (τ ))dτ ds − s

0

= I31 + I32 . As before, again an application of Lemma 5.1 (analogous result for the backward problem), (5.9), and a priori bounds for the discrete solution yield t |I31 | ≤ Ch2 s2 (vh 1 + h−1/2 ˜e¯1 )uh (s)1 ds ≤ Cth2 u0 g. 0

The term I32 is treated in a similar fashion and hence |I3 | ≤ Cth2 u0 g. For I4 and I5 , apply (3.2) and a priori estimates to have

|I4 | + |I5 | ≤ Cth

1/2

t

1/2

t

vh 21 ds

2

2

s

0

uhs 21 ds

≤ Cth2 u0 g.

0

It now remains to estimate the term I1 . We ﬁrst rewrite I1 as t t t t s(e(s), v)ds − s(e(s), e¯(s))ds + I1 = s(u, e¯(s))ds − s(Ih∗ uh − uh , vh )ds 0

0

0

0

= I11 + I12 + I13 + I14 . To estimate I11 , we note that

t−h2

I11 =

t

s(e(s), v(s))ds +

s(e(s), v(s))ds = II1 + II2 . t−h2

0

For II1 , we integrate by parts and use the fact that eˆ(0) = 0 to have

t−h2

s(ˆ es , v)ds = (t − h2 )(ˆ e(t − h2 ), v(t − h2 ))

II1 = 0

−

t−h2

(ˆ e, v)ds − 0

t−h2

s(ˆ e, vs )ds, 0

SEMIDISCRETE FINITE VOLUME ELEMENT METHOD

2341

and hence, by the Cauchy–Schwarz inequality, Lemma 5.4, and Lemma 2.3 (with time reversed), we obtain

t−h2

|II1 | ≤ tˆ e(t − h )v(t − h ) + 2

t−h2

ˆ e(s)v(s)ds +

2

0

sˆ e(s)vs (s)ds 0

t−h2 1 ≤ Cth2 u0 g + Cth2 u0 g ds (t − s) 0 (5.11) ≤ Cth2 u0 g + Cth2 ln hu0 g ≤ Cth2 ln hu0 g. By Lemma 2.3, its semidiscrete analogue, and further using a priori estimates for the backward solution v, we obtain

t

|II2 | ≤ Ct

e(s)v(s)ds ≤ Cth2 u0 g, t−h2

which together with (5.11) yields |I11 | ≤ Cth2 ln hu0 g. Since ˜e¯(t) = 0, integrate I12 by parts to have

t

I12 = −

t

s(e, ˜e¯s )ds =

t

(e, ˜e¯)ds +

0

0

s(es , ˜e¯(s))ds. 0

Apply the Cauchy–Schwarz inequality, (5.9), and a priori estimates in Lemma 2.3 to obtain

t

|I12 | ≤

t

e˜e¯(s)ds +

ses ˜e¯ds ≤ Cth2 u0 g.

0

0

Similarly, using (5.9) and Lemma 2.3 we estimate I13 as |I13 | ≤

t

˜e¯uds + 0

t

s˜e¯us ds ≤ Cth2 u0 g. 0

Finally, for I14 , apply (3.2) and a priori estimates to have

|I14 |

≤ Cth

1/2

t

uh 21

2 0

1/2

t

vh 21 ds

≤ Cth2 u0 g.

0

Altogether these estimates yield the desired result and this completes the proof. Acknowledgments. The authors wish to thank the referees for their valuable comments and suggestions which led to the improvement of the presentation of this paper. This ﬁrst author thanks the Institute for Scientiﬁc Computation and the Department of Mathematics, at Texas A&M University for providing necessary facilities for carrying out the research.

2342

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