Dynamics of Nonlinear SystemsView this Special Issue
Discontinuous Mixed Covolume Methods for Parabolic Problems
We present the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume schemes for parabolic problems on triangular meshes. We give the error analysis of the discontinuous mixed covolume schemes and obtain optimal order error estimates in discontinuous and first-order error estimate in .
The study of discontinuous Galerkin methods has been a very active research area since its introduction in  in 1973. The discontinuous Galerkin method does not require continuity of the approximation functions across the interelement boundary but instead enforces the connection between elements by adding a penalty term. Because of the use of discontinuous functions, discontinuous Galerkin methods have the advantages of a high order of accuracy, high parallelizability, localizability, and easy handling of complicated geometries. Discontinuous Galerkin methods have been used to solve hyperbolic and elliptic equations by many researchers. For example, see [2–10]. In , the unified analysis of discontinuous Galerkin methods for elliptic problems was presented. In [12, 13], Ye developed a new discontinuous finite volume method for elliptic and Stokes problems, respectively. The discontinuous finite volume method was used for parabolic equations by Bi and Geng in . In , Yang and Jiang extended a new discontinuous mixed covolume method for elliptic problems. In this paper, we consider the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume methods for the second-order parabolic problems and derive the optimal order error estimates in the discontinuous and first-order -error estimates in a mesh-dependent norm.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and describe the discontinuous mixed covolume schemes for the second-order parabolic problems and give some lemmas which will be used in the convergence analysis. In Section 3, we prove the existence and uniqueness for the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume approximation. A discontinuous mixed covolume elliptic projection is defined in Section 4. Error estimations in both discontinuous and norms of semidiscrete method and fully discrete method are proved in Sections 5 and 6.
Throughout this paper, 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
In this paper, we consider the following parabolic problems: where is a bounded convex polygonal domain with the boundary , , is an unknown function, and is a symmetric, bounded matrix function which satisfies the following condition: there exist two positive constants , such that is a given function in . Furthermore, we assume that the matrix is locally Lipschitz.
Here and in what follows, we will not write the independent , for any functions unless it is necessary.
Let , 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 . 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. 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 (3) 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 , (10) becomes Then, system (9) can be rewritten as follows:
Let . Define a mapping as where is the length of the edge . For , is defined as Then the system (13) is equivalent to Let
Using the above bilinear forms, it is clear that system (16) can be rewritten as follows:
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 (3), we have Therefore, it follows from (18) that
Let be a positive integer, let be a subdivision of time. , . We use the backward Euler difference quotient to approximate the differential quotient , 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 , one has
Lemma 2. For , one has
Lemma 3. For , one has if , then
Lemma 4. Let ; for any , there is a constant independent of such that, for is large enough,
Lemma 5. For any , there is a constant independent of such that
3. Existence and Uniqueness for Discontinuous Mixed Covolume Approximations
In this section, we prove that the discontinuous mixed covolume formulation has a unique solution in the finite element space .
Theorem 6. Semidiscrete discontinuous mixed covolume scheme (22) has a unique solution in the space .
Proof. Only prove that homogenous equation
of (22) exists unique zero solution since the number of unknowns is the same as the number of line equations.
By letting in the first formula of (32) and in the second formula of (32), using Lemma 2, the sum of (32) gives Using and Lemma 4, we have Integrating the above formula, we get Then , . So, , , . This completes the proof.
Theorem 7. The fully discrete discontinuous mixed covolume method defined in (24) has a unique solution in the finite element space .
Proof. Only prove that homogenous equation
of (24) exists unique zero solution since the number of unknowns is the same as the number of line equations.
By letting in the first formula of (36) and in the second formula of (36), using Lemma 2, the sum of (36) gives Using Lemma 4 and we have, from (37), Adding the above inequality with from to , using , 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 .
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 (22) from those of (21), respectively, we have
Using (46), we have
Differentiating the first equation of (51) on , we have
By letting in the second formula of (51) and letting in (52), using Lemma 2, the sum of them gives Using we have Multiplying the equation above with 2, integrating them from 0 to , and using -inequality, Lemma 4, and (48), we can get then so hence
Now, using the triangle inequalities (47), (58), and (60), we get The proof is complete.
6. Error Estimates for Fully Discrete Method
Let , , ; then the error estimates for the backward Euler fully discrete discontinuous mixed covolume method in the and norms are provided in the next two theorems.
Proof. Subtracting the two equations of (24) from (21), respectively, with , we can get the error equation:
Choosing and in the two equations of (63), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection, we have
First, we estimate the left item of (64). Using Lemma 4, we have
Then, we estimate the right item of (64). From we have therefore Substituting the estimations above into (64), using , we have By -inequality and the discrete Gronwall inequality, we have From the above formula and (47) and using the triangle inequality, we have This completes the proof.
Proof. Choosing and in the two equations of (63), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection, we have From and (66) and (67), we have therefore From the above formula and (47) and using the triangle inequality, we have The proof is complete.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project is supported by the fund of the National Natural Science (11171193), the fund of the Natural Science of Shandong Province (ZR2011MA016), and a Project of Shandong Province Science and Technology Development Program (2012GGB01198).
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