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 H(div) and first-order error estimate in L 2.


Introduction
The study of discontinuous Galerkin methods has been a very active research area since its introduction in [1] 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][3][4][5][6][7][8][9][10]. In [11], 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 [14]. In [15], 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 (div) and first-order 2 -error estimates in a meshdependent 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 (div) and 2 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.

Discontinuous Mixed Covolume Formulation
In this paper, we consider the following parabolic problems: where Ω ∈ R 2 is a bounded convex polygonal domain with the boundary Ω, x = ( , ), is an unknown function, and is a symmetric, bounded matrix function which satisfies 2 The Scientific World Journal the following condition: there exist two positive constants 1 , is a given function in 2 (Ω). Furthermore, we assume that the matrix = −1 is locally Lipschitz.
Here and in what follows, we will not write the independent x, for any functions unless it is necessary.
Let T ℎ = { } be a triangulation of the domain Ω. As usual, we assume the triangles to be shape-regular. For a given triangulation T ℎ , we construct a dual mesh T * ℎ based upon the primal partition T ℎ . Each triangle in T ℎ can be divided into three subtriangles by connecting the barycenter of the triangle to their corner nodes ( = 1, 2, 3). Then, we define the dual partition T * ℎ 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 T ℎ by Define the finite-dimensional test function space ℎ for velocity associated with the dual partition T * ℎ as Let ℎ be the finite-dimensional space for pressure: Let Γ denote the union of the boundary of the triangles of T ℎ and Γ 0 : Γ \ Ω. The traces of functions in ℎ and ℎ are double valued on Γ 0 . Let be an interior edge shared by two triangles 1 and 2 in ℎ . Define the normal vectors n 1 and n 2 on pointing exterior to 1 and 2 , 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 v, respectively, If is an edge on the boundary of Ω, we set where n is the outward unit normal. We do not require either of the quantities [ ] or {k} on boundary edges, and we leave them undefined.

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 ℎ × ℎ .

Theorem 7.
The fully discrete discontinuous mixed covolume method defined in (24) has a unique solution in the finite element space ℎ × ℎ .
Choosing w = ( = 1, 2, . . . , ) and = ( = 1, 2, . . . , ) in the two equations of (63), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection, we have Then, we estimate the right item of (64). From we have Substituting the estimations above into (64), using 0 = 0, we have The Scientific World Journal 7 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.