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 H(div) and first-order error estimate in L2 are obtained with the lowest order Raviart-Thomas mixed element space.

Here and in what follows, we will not write the independent x,  for any functions unless it is necessary.
For the parabolic integrodifferential problems many numerical methods were proposed, such as the finite element methods in [1], H1-Galerkin mixed finite element methods in [2], finite element approximation with a weakly singular kernel in [3], expanded mixed finite element methods in [4], and expanded mixed covolume method in [5].
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 convectiondiffusion problems by many researchers; see [6][7][8][9][10][11].The discontinuous finite volume method in recent years was used to solve elliptic problems, Stokes problems, and parabolic problems in [12][13][14].In [15] 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 [16].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  1 and  2 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.
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 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 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 k, 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 {u} on boundary edges, and we leave them undefined.

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  ℎ ×  ℎ .
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 k = w ℎ in the first formula of (30) and  =  ℎ in the second formula of (30), using Lemma 2, the sum of (30) gives Using Hölder inequality and Gronwall Lemma, we get Integrating the above formula, we get Then ‖ ℎ ‖ = 0, |‖w ℎ ‖| div = 0.So  ℎ = 0, w ℎ = 0,  ∈ (0, ].This completes the proof. of ( 23) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting k = w  ℎ in the first formula of (35) and  =   ℎ in the second formula of (35), using Lemma 2, the sum of (35) gives Using Lemmas 3 and 4 and
Adding the above inequality with  from 1 to , using  0 ℎ = 0 and the discrete Gronwall inequality, when Δ is sufficiently small, we have

A Discontinuous Mixed Covolume Elliptic Projection
Define an operator   from  1 () to  1 () by requiring that, for any ∀ ∈  1 (), where   ( = 1, 2, 3) are the three sides of the element  ∈ T ℎ .It was proved in [5] that For any u ∈  1 0 (Ω) 2 , define Π 1 u ∈  ℎ by Using the definition of Π 1 and integration by parts, we can show that It was proved in [6] that Let Π 2 be the projection from  2 0 (Ω) to the finite element space  ℎ .

Error Estimates for Semidiscrete Method
In this section, we will establish the error estimates in the (div) and  2 norms for the semidiscrete discontinuous mixed covolume method.Theorem 9. Let (w ℎ ,  ℎ ) ∈  ℎ ×  ℎ be the solution of (21) and  ℎ (0) = qℎ (0), (w, ) ∈  2 (Ω) 2 ×  1 (Ω) the solution of (2) Differentiating the first equation of (50) on , we have that By letting  =   in the second formula of (50) and letting k =  in (51), using Lemma 2, the sum of them gives and Lemmas 1 and 3 gives The proof is complete.