Journal of Mathematics

Volume 2013 (2013), Article ID 464893, 13 pages

http://dx.doi.org/10.1155/2013/464893

Research Article

On the Finite Volume Element Method for Self-Adjoint Parabolic Integrodifferential Equations

Mohamed Bahaj¹ and Anas Rachid²

¹Department of Mathematics and Computing Science, Faculty of Sciences and Technology, Hassan 1st University, BP 577 Settat, Morocco

²École Nationale Suéprieure d’Arts et Métiers-Casablanca, Université Hassan II Mohammedia-Casablanca, BP 150 Mohammedia, Morocco

Received 26 December 2012; Accepted 15 April 2013

Academic Editor: Mario Ohlberger

Copyright © 2013 Mohamed Bahaj and Anas Rachid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Finite volume element schemes for non-self-adjoint parabolic integrodifferential equations are derived and stated. For the spatially discrete scheme, optimal-order error estimates in , , and , norms for are obtained. In this paper, we also study the lumped mass modification. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.

1. Introduction

The main purpose of this paper is to study semidiscrete and full discrete finite volume element method (FVE) for parabolic integrodifferential equation of the form

where is a bounded domain in ,  , with smooth boundary , and . Here , a non-self-adjoint second-order strongly elliptic, and , an arbitrary second-order linear partial differential operator, both with coefficients depending smoothly on and ,   and are known functions, which are assumed to be smooth and satisfy certain compatibility conditions for and , so that (1) has a unique solution in certain Sobolev space. Problem (1) occurs in nonlocal reactive flows in porous media, viscoelasticity, and heat conduction through materials with memory.

Finite volume method is an important numerical tool for solving partial differential equations. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer, and petroleum engineering. The method can be formulated in the finite difference framework or in the Petrov-Galerkin framework. Usually, the former one is called finite volume method [1], marker and cell (MAC) method [2], or cell-centered method [3], and the latter one is called finite volume element method (FVE) [4–9], covolume method [10], or vertex-centered method [11, 12]. We refer to the monographs [13, 14] for general presentation of these methods. The most important property of FVE method is that it can preserve the conservation laws (mass, momentum, and heat flux) on each control volume. This important property, combined with adequate accuracy and ease of implementation, has attracted more people to do research in this field.

Recently, the authors in [8, 15] studied FVE method for general self-adjoint elliptic problems. The authors in [16] presented and analyzed the semidiscrete and full discrete symmetric finite volume schemes for a class of parabolic problems. In [6, 7] the authors have studied FVE for one- and two-dimensional parabolic integrodifferential equations and have obtained an optimal-order estimate in the -norm. The regularity required on the exact solution is for which is higher when compared to that for finite element methods.

The aim of this paper is to study the convergence of FVE discretization for a nonself-adjoint parabolic integrodifferential problem (1). Both spatially discrete scheme and discrete-in-time scheme are analyzed, and optimal error estimates in and norms are proved using only energy method. We also explore and generalize that idea to develop the lumped mass modification and estimates, . Our analysis avoids the use of semigroup theory, and the regularity requirement on the solution is the same of that of finite element method. Furthermore, based on the Crank-Nicolson method the fully discrete scheme is analyzed and the related optimal error estimates are established.

This paper is organized as follows. In Section 2, we introduce some notations and present some preliminary materials to be used later. The Ritz-Volterra projection to finite volume element spaces is introduced and related estimates are carried out in Section 3. In Section 4, we estimate the error of the finite volume element approximations derived in the previous section. In Section 5, the lumped mass is presented and optimal estimates in and norms are obtained Finally, the Crank-Nicolson scheme is studied in Section 6.

2. Finite Volume Element Scheme

In this section, we introduce some material which will be used repeatedly hareafter. Throughout this paper, (with or without index) denotes a generic positive constant which does not depend on the spatial and time discretization parameters and ,  respectively.

2.1. Notations

We will use and (resp., and ) to denote the norm and seminorm of the Sobolev space (resp., ). The scalar product and norm in are denoted by and , respectively. Let be the standard Sobolev subspace of of functions vanishing on .

The weak form of (1) is used to find ,  such that

where

Let be a decomposition of into triangles (for the 2D case) or tetrahedral (for the 3D case) with , where is the diameter of the element .

In order to describe the FVE method for solving problem (1), we will introduce a dual partition based upon the original partition whose elements are called control volumes. We construct the control volumes in the same way as in [7, 17]. Let be a point of . In the 2D case, on each edge of a point is selected; then we connect with line segments to ; thus, partitioning into three quadrilaterals ,  , where are the vertices of . Then with each vertex we associate a control volume , which consists of the union of the subregions , sharing the vertex    (see Figure 1).

(a)

(b)

Figure 1: (a) A sample region with blue lines indicating the corresponding control volume . (b) A triangle partitioned into three subregions .

Similarly, in the 3D case, on each of the four faces , a point , is selected, and on each of the six edges a point is selected. On each of the two faces and of sharing an edge , we connect ,  with and with by line segments, thus, partitioning into twelve tetrahedron ,   (see Figure 2). Then for the control volume consists of the union of the subregions sharing the vertex . Thus, we finally obtain a group of control volumes covering the domain , which is called the dual partition of the triangulation . We denote by the set of interior vertices and . For a vertex ,  let be the index set of those vertices that along with are in some element of (Figure 2).

Figure 2: A tetrahedron partitioned into twelve subregions .

There are various ways to introduce a regular dual partition . In this paper, we will also use the construction of the control volumes in which we let be the barycenter of . In the 2D case, we choose to be the midpoint of the edge (see Figure 3).

(a)

(b)

Figure 3: is the barycenter of , and is to be the midpoint of the edge .

In the 3D case, we choose to be the midpoint of the edge and to be the barycenter of the face   (Figure 4).

Figure 4: is the midpoint of the edge , and   is the barycenter of the face .

We call the partition regular or quasiuniform, if there exists a positive such that

If the finite element triangulation is quasiuniform, that is, there exists a positive such that

then the dual partitionis also quasiuniform.

Based on the triangulation, let be the standard conforming finite element space of piecewise linear functions, defined on the triangulation as follows:

Let be the standard interpolation operators, such that

where are the standard basis functions of and .

2.2. Construction of the FVE Scheme

We formulate the FVE method for the problem (1) as follows: Given a , integrating (1)¹ over the associated control volume and applying Green’s formula, we obtain an integral conservation as follows form:

wheredenotes the unit outer normal vector to .

Let be the transfer operator defined by

where

and is the characteristic function of the control volume .

Now for and for an arbitrary , we multiply (8) by , and sum over all . Then the semidiscrete FVE approximation of (1) is a solution to the following problem: find for such that

Here the bilinear forms and are defined by

Let

Then, we can rewrite scheme (11)¹ as systems of ordinary differential equations as follows:

Here ,  the mass matrix is tridiagonal, and both and are positive definites.

In order to describe features of the bilinear forms defined in (11), we introduce some discrete norms on in the same way as in [7]:

where ,  the distance between and . Obviously, these norms are well defined for as well and .

Hereafter we state the equivalence of discrete norms and with usual norms and on , respectively.

Lemma 1 (see [7]). There exist two positive constants and such that for all , we have

Next we recall some properties of the bilinear forms (see [7, 18])

Lemma 2 (see [7]). There exist two positive constants and such that for all ,  we have

The following lemmas are proved in [3, 7], which give the key feature of the bilinear forms in the FVE method.

Lemma 3 (see [3]). Assume that . Then, one has

The aforementioned identity holds true when is replaced by .

Lemma 4 (see [3]). Assume that . Then, one has

Furthermore, for ,  we have

3. Ritz-Volterra Projection and Related Estimates

Following [7, 19, 20], we define the Ritz-Volterra projection as follows:

This is an elliptic projection with memory of into . It is easy to see that (21) is actually a system of integral equations of Volterra type. In fact, if , then (21) can be rewritten as

where ,   are matrices and ,   are vectors, defined via

From the positivity of (Lemma 2) and the linearity of (22), we see that the system (22) possesses a unique solution . Consequently, in (21) is well defined.

Set . The following lemma was proved in [7], which shows the error estimate for and its temporal derivative.

Lemma 5 (see [7]). Assume that for all , for some integer .  Then, for fixed there is a constant , independent of and , such that for all and ,

Now we establish error estimate for and its temporal derivative which improves Theorem  2.2 in [7]. This estimate is optimal with respect to the order.

Lemma 6. Assume that, for some integer ,    for all . Then, for fixed there is a constant , independent of and , such that for all and ,

Proof. The proof will proceed by duality argument. Let be the solution of

The solution satisfies the following regularity estimate:

Multiplying this equation by and then taking innerproduct over , we obtain the following:

We have

Applying Lemma 4, we obtain

Finally, we have

then, we have

Finally, an application of Gronwall’s lemma yields the first estimate.

The second inequality follows in a similar fashion.

Lemma 7. There exists a constant independent of such that

Proof. Let be an arbitrary component of , with and conjugate indices; we have .

For any such , let be the solution of

It follows from the regularity theory for the elliptic problem that

We then have by application of (21) that

Applying Lemma 4, we have

Finally, is estimated as follows:

Combining these estimates, we get

hence by Gronwall’s lemma

The derivation of the error estimate in is similar to the case when .

4. Error Estimates for Semidiscrete Approximations

We split the error as follows:

It is easy to see that satisfies an error equation of the form

Since the estimates of are already known, it is enough to have estimates for .

We will prove a sequence of lemmas which lead to the following result.

Lemma 8. There is a positive constant independent of such that

Proof. Since we may take in (42) to obtain

and hence by integration and Lemma 1, we have

Gronwall’s lemma now implies the following:

Since this holds for all , we may conclude that

Remark 9. If the initial value was chosen so that ,  then . One can derive

Lemma 10. There is a positive constant independent of such that

Proof. Set in (42) to get

Then

In addition, recall that

then applying an inverse inequality and using kickback argument, we obtain

Combining these estimates, we derive

So after integration in time and using the weak coercivity of , we get

and by Gronwall’s lemma,

Remark 11. If , then

Theorem 12 (error estimates in and -norms). Let ,   be the solutions of (2) and (11), respectively. Assume that .

(a)Let be chosen so that . Then for fixed there is a constant independent of , such that for all ,

(b)Let be chosen so that . Then for fixed there is a constant independent of , such that for all ,

We now prove error estimates for FVE approximations in and -norms.

Theorem 13 (error estimates inand-norms). Let ,   be the solutions of (2) and (11), respectively and . Assume that . For sufficiently small, we have

Proof. If , by the following Sobolev embedding inequality

then the first desired estimate follows from Lemmas 7 and 10.

Given , find such that

We have

By Lemma 4

where we have used the fact ,  . Combining these estimates, we get

Hence using the Poincaré inequality, we have forsufficiently small

We compare the relationship between covolume solution and the Galerkin finite element solution.

Corollary 14. Letbe the finite element solution to (2); that is,

For sufficiently small, we have

Proof. By (2) and (67),

Consider the following auxiliary problem. For any such , let be the solution of the following:

with

On the other hand,

where we have used the fact ,  

We deduce the result from the known finite element estimates.

Remark 15. In order to estimate , by differentiating (42) with respect to , we obtain

Setting , we obtain

Using kickback argument, integrating and applying Gronwall’s lemma, we deduce

5. The Lumped Mass Finite Volume Element Method

In this section, we restrict our study to the 2D case. A simple way to define the lumped mass scheme [21] is to replace the mass matrix in (14) by the diagonal matrix obtained by taking for its diagonal elements the numbers or by lumping all masses in one row into the diagonal entry. This makes the inversion of the matrix in front of a triviality. We will therefore study the matrix problem

We know that the lumped mass method defined by (77) above is equivalent to

Our alternative interpretation of this procedure will be to think of (77) as being obtained by evaluating the first term in (78) by numerical quadrature. Let be a triangle of the triangulation , let ,  be its vertices, and consider the quadrature formula

We may then define the associated bilinear form in, using the quadrature scheme, by the following:

We note that is a norm in which is equivalent to the -norm uniformly in ; that is, there exist two positive constants and such that for all , we have

We note that the aforementioned definition may be used also for   and that for .

The lumped mass method defined by (78) is equivalent to

We introduce the quadrature error

Lemma 16 (see [21]). Let . Then

Theorem 17. Let and be the solutions of (82) and (2), respectively, and assume . Then we have for the error in the lumped mass semidiscrete method, for , the following:

Proof. In order to estimate , we write

We rewrite

Setting in (87), we obtain

Using Lemma 16 and the inverse estimate, we get

we have

Using Young’s inequality and Gronwall’s lemma to eliminate on the right-hand side and using integration in , we get the result.

Using Young’s inequality to eliminate on the right hand side it becomes.

Using integration in , we get the result.

We will now show that the -norm error bound of theorem remains valid for the lumped mass method (82).

Theorem 18. Let and be the solutions of (82) and (2), respectively, and assume

Then, we have for the error in the lumped mass semidiscrete method, for , the following:

Proof. Setting in (87), we obtain

It follows, thus, that using integration in and Gronwall’s lemma, we have

6. Full Discretization

Let be the backward difference quotient of ; assume that is a discrete analogue of (similarly ), where the projection is defined by

In order to define fully discrete approximation of (11), we discretize the time by taking ,  ,     and use the numerical quadrature

Here are the integration weights and we assume that the following error estimate is valid:

Now, define our complete discrete FVE approximation of (11) by the following: find for , such that for all

where .

Theorem 19. Let and be the solutions of problem (2) and its complete discrete scheme (99), respectively. Then for any there exists a positive constant , independent of , such that for

Proof. Let us split the error into two parts: ,  where   and , and let be the Ritz-Volterra projection of . Then from (2) and (99) we have for all the following:

where

In fact, by Taylor expansion,

we have

In addition, the quadrature error satisfies

Taking in (101) and noting that , there is

Summing from to and then, after cancelling the common factor and using Gronwall’s lemma, we obtain

and then

the theorem follows from the estimates of and .

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