Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems
The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuous and first-order error estimate in are obtained with the lowest order Raviart-Thomas mixed element space.
We consider the following linear parabolic integrodifferential problems where is a bounded convex polygonal domain with the boundary , , is an unknown function, is a symmetric, bounded matrix function, is a bounded matrix function, and are known functions, and . Furthermore, we assume that the matrix and are two bounded matrix functions.
Here and in what follows, we will not write the independent , for any functions unless it is necessary.
For the parabolic integrodifferential problems many numerical methods were proposed, such as the finite element methods in , H1-Galerkin mixed finite element methods in , finite element approximation with a weakly singular kernel in , expanded mixed finite element methods in , and expanded mixed covolume method in .
Because the discontinuous Galerkin method has the advantages of a high order of accuracy, high parallelizability, localizability, and easy handling of complicated geometries, it has been used to solve elliptic problems and convection-diffusion problems by many researchers; see [6–11]. The discontinuous finite volume method in recent years was used to solve elliptic problems, Stokes problems, and parabolic problems in [12–14]. In  the discontinuous mixed covolume methods for elliptic problems were demonstrated by Yang and Jiang. Zhu and Jiang extended the discontinuous mixed covolume methods to parabolic problems in . The goal of this paper is to extend the discontinuous mixed covolume methods in the linear parabolic integrodifferential problems.
The rest of this paper is organized as follows. In Section 2, some notations are introduced and the semidiscrete and the fully discrete discontinuous mixed covolume schemes for the integrodifferential equations (1) are established. In Section 3, the existence and uniqueness for the semidiscrete and the fully discrete discontinuous mixed covolume approximations are proven. We defined a generalized discontinuous mixed covolume elliptic projection in Section 4. We prove the optimal error estimations in both and norms of semidiscrete and the fully discrete discontinuous mixed covolume methods in Sections 5 and 6.
Throughout this paper, the letter denotes a generic positive constant independent of the mesh parameter and may stand for different values at its different appearances.
2. Discontinuous Mixed Covolume Formulation
Let and , and rewrite (1) as the system of first-order partial differential equations
We will use the standard definitions for the Sobolev spaces and their associated inner products , norms and seminorms in . The space coincides with , in which the norm and the inner product are denoted by and , respectively.
Let be a triangulation of the domain . As usual, we assume the triangles to be shape-regular. For a given triangulation , we construct a dual mesh based upon the primal partition . Each triangle in can be divided into three subtriangles by connecting the barycenter of the triangle to their corner nodes . Then we define the dual partition to be the union of the triangles shown in Figure 1. Let consist of all the polynomials functions of degree less than or equal to defined on . We define the finite-dimensional trial function space for velocity on by Define the finite-dimensional test function space for velocity associated with the dual partition as Let be the finite-dimensional space for pressure
Let denote the union of the boundary of the triangles of and . The traces of functions in and are double valued on . Let be an interior edge shared by two triangles and in . Define the normal vectors and on pointing exterior to and , respectively. Next, we introduce some traces operators that we will use in our numerical formulation. We define the average and jump on for scalar and vector , respectively: if is an edge on the boundary of , we set where is the outward unit normal. We do not require either of the quantities or on boundary edges, and we leave them undefined.
Multiplying the first and second equations in system (2) by and , respectively, and using the integration by parts formula in the equation, we have where is the outward normal vector on . Let be the triangles in . Then we have where . A straightforward computation gives Let . Using the above formula and the fact that for on , (9) becomes Then, system (8) can be rewritten as in the following:
Let . Define a mapping as where is the length of the edge . For , is defined as Then the system (12) is equivalent to Let Using the above bilinear forms, it is clear that system (15) can be rewritten as in the following:
In order to define our numerical schemes, we introduce the bilinear forms as follows: where is a parameter to be determined later. For the exact solution of system (2), we have Therefore, it follows from (17) that
Let be a positive integer; let be a subdivision of time. . We use the backward Euler difference quotient to approximate the differential quotient and the numerical integration to approximate the integration in the semidiscrete scheme; then we obtain the backward Euler fully discrete discontinuous mixed covolume scheme for the problem (1): find , such that where , , , will be given in Section 4.
We define the following norms for : where is the function whose restriction to each element is equal to and .
We will introduce some useful Lemmas; for more details, see .
Lemma 1. For , we have
Lemma 2. For , we have
Lemma 3. For , we have if , then
Lemma 4. Let ; for any , there is a constant independent of such that, for is large enough,
3. Existence and Uniqueness for Discontinuous Mixed Covolume Approximations
In this section, we prove the discontinuous mixed covolume formulation has a unique solution in the finite element space .
Theorem 5. The semidiscrete discontinuous mixed covolume scheme (21) has a unique solution in the space .
Proof. Only prove that homogenous equation
of (21) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting in the first formula of (30) and in the second formula of (30), using Lemma 2, the sum of (30) gives Using and Lemmas 3 and 4, we have that Using Hölder inequality and Gronwall Lemma, we get Integrating the above formula, we get Then , . So , , . This completes the proof.
Theorem 6. The fully discrete discontinuous mixed covolume method defined in (23) has a unique solution in the finite element space if is sufficiently small.
Proof. Only prove that homogenous equation
of (23) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting in the first formula of (35) and in the second formula of (35), using Lemma 2, the sum of (35) gives Using Lemmas 3 and 4 and we have from (36) that Adding the above inequality with from 1 to , using and the discrete Gronwall inequality, when is sufficiently small, we have Hence we have and ; that is, and . This completes the proof.
4. A Discontinuous Mixed Covolume Elliptic Projection
Define an operator from to by requiring that, for any , where are the three sides of the element . It was proved in  that
For any , define by Using the definition of and integration by parts, we can show that It was proved in  that Let be the projection from to the finite element space .
Define a discontinuous mixed covolume elliptic projection by requiring that, finding : , such that It was proved in  that the above formula has a unique solution and the error estimates in the following Theorem 7.
5. Error Estimates for Semidiscrete Method
In this section, we will establish the error estimates in the and norms for the semidiscrete discontinuous mixed covolume method.
Proof. Let , . Subtracting the two equations of (21) from those of (20), respectively, we have
Using (45), we have
Differentiating the first equation of (50) on , we have that
By letting in the second formula of (50) and letting in (51), using Lemma 2, the sum of them gives Using and Lemmas 1 and 3 gives Multiplying the equation above with 2, integrating them from 0 to and using Hölder inequality, -inequality, Gronwall inequality, Lemma 4, and (47), we can get so hence
Now, using the triangle inequality, (46), (56), and (58), we get The proof is complete.
6. Error Estimates for Fully Discrete Method
Let , , , and then the error estimates for the backward Euler fully discrete discontinuous mixed covolume method in the and norms are provided in next two theorems.
Proof. Subtracting the two equations of (23) from (20), respectively, with , we can get the error equation:
Choosing and in the two equations of (61), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection with , we have
First, we estimate the left item of (62). Using Lemma 4, we have
Then, we estimate the right item of (62). From we have and therefore Using Lemma 3, we can get
Substituting the estimations above into (62), multiplying them with , adding them with from 1 to , and using , we have Using the discrete Gronwall inequality, we have From the formula above and (46) and using the triangle inequality, we have This completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author acknowledges a project supported by the fund of National Natural Science (11171193) and a project of Shandong Province Science and Technology Development Program (2012GGB01198).
A. Zhu and Q. Yang, “Expanded mixed finite element methods for parabolic integro-differential equations,” Journal of Shandong Normal University, vol. 19, no. 2, pp. 10–14, 2004.View at: Google Scholar
W. H. Reed and T. R. Hill, “Triangular mesh methods for the neutron transport equation,” Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, USA, 1973.View at: Google Scholar
B. Cockburn, G. E. Karniaddakis, and C. W. Shu, The Development of Discontinuous Galerkin Methods, Springer, Berlin, Germany, 2000.
R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975.View at: MathSciNet