Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems

Abstract

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.

1. Introduction

We consider the following linear parabolic integrodifferential problems 𝑞_𝑡(𝐱,𝑡)−∇⋅𝐴(𝐱,𝑡)∇𝑞(𝐱,𝑡)+^𝑡₀𝐵()()(),()(],𝑞()𝐱,𝑡,𝜏∇𝑞𝐱,𝑡,𝜏𝑑𝜏=𝑓𝐱,𝑡𝐱,𝑡∈Ω×0,𝑇𝐱,0=𝑞₀(𝐱)𝑞()()(],,𝐱∈Ω,𝐱,𝑡=0,𝐱,𝑡∈𝜕Ω×0,𝑇 (1) where Ω∈𝐑² is a bounded convex polygonal domain with the boundary 𝜕Ω, 𝐱=(𝑥,𝑦), 𝑞 is an unknown function, 𝐴 is a symmetric, bounded matrix function, 𝐵 is a bounded matrix function, 𝑞₀(𝐱) and 𝑓(𝐱,𝑡) are known functions, and 𝑓(𝐱,𝑡)∈𝐿²(Ω). Furthermore, we assume that the matrix 𝑀=𝐴⁻¹ and 𝑀_𝑡 are two bounded matrix functions.

Here and in what follows, we will not write the independent 𝐱, 𝑡 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 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 [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 𝐻¹ and 𝐿² norms of semidiscrete and the fully discrete discontinuous mixed covolume methods in Sections 5 and 6.

Throughout this paper, the letter 𝐶 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 𝐰=−𝐴∇𝑞 and 𝐷=𝐵𝑀, and rewrite (1) as the system of first-order partial differential equations ()(],𝑞𝑀𝐰+∇𝑞=0,𝐱,𝑡∈Ω×0,𝑇_𝑡+∇⋅𝐰+^𝑡₀()()(],𝑞()∇⋅𝐷𝐰𝑑𝜏=𝑓,𝐱,𝑡∈Ω×0,𝑇𝐱,0=𝑞₀(𝐱)𝑞()()(].,𝐱∈Ω,𝐱,𝑡=0,𝐱,𝑡∈𝜕Ω×0,𝑇 (2)

We will use the standard definitions for the Sobolev spaces 𝐻^𝑠(𝐾) and their associated inner products (⋅,⋅)_𝑠,𝐾, norms ‖⋅‖_𝑠,𝐾 and seminorms |⋅|_𝑠,𝐾 in [17]. The space 𝐻⁰(𝐾) coincides with 𝐿^𝑠(𝐾), in which the norm and the inner product are denoted by ‖⋅‖_𝐾 and (⋅,⋅)_𝐾, respectively.

Let 𝒯_ℎ={𝐾} be a triangulation of the domain Ω. As usual, we assume the triangles 𝐾 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 𝑄 of the triangle to their corner nodes 𝐴_𝑖(𝑖=1,2,3). Then we define the dual partition 𝒯^∗_ℎ to be the union of the triangles shown in Figure 1. Let 𝑃_𝑘(𝑇) consist of all the polynomials functions of degree less than or equal to 𝑘 defined on 𝑇. We define the finite-dimensional trial function space for velocity on 𝒯_ℎ by 𝑈_ℎ∶=𝐮∈𝐿²(Ω)²∶𝐮|_𝐾∈𝑃₁(𝐾)²,∀𝐾∈𝒯_ℎ. (3) Define the finite-dimensional test function space 𝑉_ℎ for velocity associated with the dual partition 𝒯^∗_ℎ as 𝑉_ℎ∶=𝐯∈𝐿²(Ω)²∶𝐯|_𝑇∈𝑃₀(𝑇)²,∀𝑇∈𝒯^∗_ℎ. (4) Let 𝐻_ℎ be the finite-dimensional space for pressure 𝐻_ℎ∶=𝑝∈𝐿²(Ω)∶𝑝_𝐾∈𝑃₀(𝐾),∀𝐾∈𝒯_ℎ. (5)

Figure 1 

Element 𝑇∈𝒯∗ℎ for a triangular mesh.

Let Γ denote the union of the boundary of the triangles 𝐾 of 𝒯_ℎ and Γ⁰∶Γ⧵𝜕Ω. The traces of functions in 𝑉_ℎ and 𝐻_ℎ are double valued on Γ⁰. Let 𝑒 be an interior edge shared by two triangles 𝐾₁ and 𝐾₂ in 𝑇_ℎ. Define the normal vectors 𝐧₁ and 𝐧₂ on 𝑒 pointing exterior to 𝐾₁ and 𝐾₂, respectively. Next, we introduce some traces operators that we will use in our numerical formulation. We define the average {⋅} and jump [⋅] on 𝑒 for scalar 𝑞 and vector 𝐯, respectively: 𝑝=12𝑝_𝜕𝐾1+𝑝_𝜕𝐾2,𝑝=𝑝_𝜕𝐾1𝐧₁+𝑝_𝜕𝐾2𝐧₂,{𝐮}=12𝐮|_𝜕𝐾1+𝐮|_𝜕𝐾2,[𝐮]=𝐮|_𝜕𝐾1⋅𝐧₁+𝐮|_𝜕𝐾2⋅𝐧₂; (6) if 𝑒 is an edge on the boundary of Ω, we set 𝑝=𝑝,[𝐮]=𝐮⋅𝐧, (7) where 𝐧 is the outward unit normal. We do not require either of the quantities [𝑝] or {𝐮} on boundary edges, and we leave them undefined.

Multiplying the first and second equations in system (2) by 𝐯∈𝑉_ℎ and 𝑝∈𝐻_ℎ, respectively, and using the integration by parts formula in the equation, we have _{𝑇∈𝒯∗ℎ𝑇}𝑀𝐰⋅𝐯𝑑𝑥+_{𝑇∈𝒯∗ℎ𝜕𝑇}𝑞𝐯⋅𝐧𝑑𝑠=0,∀𝐯∈𝑉_ℎ,_{𝐾∈𝒯ℎ𝐾}𝑞_𝑡𝑝𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}∇⋅𝐰𝑝𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}^𝑡₀∇⋅(𝐷𝐰)𝑝𝑑𝜏𝑑𝑥=𝑓,𝑝,∀𝑝∈𝐻_ℎ, (8) where 𝐧 is the outward normal vector on 𝜕𝑇. Let 𝑇_𝑗∈𝒯^∗_ℎ(𝑗=1,2,3) be the triangles in 𝐾∈𝒯_ℎ. Then we have _{𝑇∈𝒯∗ℎ𝜕𝑇}𝑞𝐯⋅𝐧𝑑𝑠=_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}𝑞𝐯⋅𝐧𝑑𝑠+_{𝐾∈𝒯ℎ𝜕𝐾}𝑞𝐯⋅𝐧𝑑𝑠,∀𝐯∈𝑉_ℎ, (9) where 𝐴₄=𝐴₁. A straightforward computation gives _{𝐾∈𝒯ℎ𝜕𝐾}𝑞𝐯⋅𝐧𝑑𝑠=_{𝑒∈Γ0𝑒}𝑞⋅{𝐯}𝑑𝑠+_{𝑒∈Γ𝑒}𝑞⋅[𝐯]𝑑𝑠,∀𝐯∈𝑉_ℎ. (10) Let ∫_Γ𝑞𝑑𝑠=∑_𝑒∈Γ∫_𝑒𝑞𝑑𝑠. Using the above formula and the fact that [𝑞]=0 for 𝑞∈𝐻¹(Ω) on Γ⁰, (9) becomes _{𝑇∈𝒯∗ℎ𝜕𝑇}𝑞𝐯⋅𝐧𝑑𝑠=_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}𝑞𝐯⋅𝐧𝑑𝑠+_{𝑒∈Γ𝑒}𝑞[𝐯]𝑑𝑠,∀𝐯∈𝑉_ℎ. (11) Then, system (8) can be rewritten as in the following: _{𝑇∈𝒯∗ℎ𝑇}𝑀𝐰⋅𝐯𝑑𝑥+_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}𝑞𝐯⋅𝐧𝑑𝑠+_{𝑒∈Γ𝑒}𝑞[𝐯]𝑑𝑠=0,∀𝐯∈𝑉_ℎ,_{𝐾∈𝒯ℎ𝐾}𝑞_𝑡𝑝𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}∇⋅𝐰𝑝𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}^𝑡₀∇⋅(𝐷𝐰)𝑝𝑑𝜏𝑑𝑥=𝑓,𝑝,∀𝑝∈𝐻_ℎ. (12)

Let 𝑈(ℎ)=𝑈_ℎ+𝐻²(Ω)². Define a mapping 𝛾∶𝑈(ℎ)→𝑉_ℎ as 𝛾𝐮|_𝑇=1ℎ_𝑒𝑒𝐮|_𝑇𝑑𝑠,𝑇∈𝒯^∗_ℎ, (13) where ℎ_𝑒 is the length of the edge 𝑒. For 𝐮=(𝑢₁,𝑢₂)∈𝑈(ℎ), 𝛾𝑢_𝑖(𝑖=1,2) is defined as 𝛾𝑢_𝑖𝑇=1ℎ_𝑒𝑒𝑢_𝑖𝑇𝑑𝑠,𝑇∈𝒯^∗_ℎ,(𝑖=1,2). (14) Then the system (12) is equivalent to _{𝑇∈𝒯∗ℎ𝑇}𝑀𝐰⋅𝛾𝐯𝑑𝑥+_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}𝑞𝛾𝐯⋅𝐧𝑑𝑠+_{𝑒∈Γ𝑒}𝑞𝛾𝐯𝑑𝑠=0,∀𝐯∈𝑈_{𝐾∈𝒯ℎ𝐾}(ℎ),𝑞_𝑡𝑝𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}∇⋅𝐰𝑝𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}^𝑡₀∇⋅(𝐷𝐰)𝑝𝑑𝜏𝑑𝑥=𝑓,𝑝,∀𝑝∈𝐻_ℎ. (15) Let 𝑎₀(𝐮,𝐯)∶=_{𝑇∈𝒯∗ℎ𝑇}𝑏𝑀𝐮⋅𝛾𝐯𝑑𝑥,𝐯,𝑝∶=_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}𝑝𝛾𝐯⋅𝐧𝑑𝑠+_{𝑒∈Γ𝑒}𝑝𝑐𝛾𝐯𝑑𝑠,₀𝐮,𝑝∶=_{𝐾∈𝒯ℎ𝐾}𝑐̃∇⋅𝐮𝑝𝑑𝑥,₀𝐮,𝑝∶=_{𝐾∈𝒯ℎ𝐾}∇⋅(𝐷𝐮)𝑝𝑑𝑥. (16) Using the above bilinear forms, it is clear that system (15) can be rewritten as in the following: 𝑎₀()(ℎ),𝑞𝐰,𝐯+𝑏𝐯,𝑞=0,∀𝐯∈𝑈_𝑡,𝑝+𝑐₀𝐰,𝑝+^𝑡₀𝑐₀̃𝐰,𝑝𝑑𝜏=𝑓,𝑝,∀𝑝∈𝐻_ℎ. (17)

In order to define our numerical schemes, we introduce the bilinear forms as follows: 𝑎(𝐮,𝐯)∶=𝑎₀(𝐮,𝐯)+𝛼_𝑒∈Γ1ℎ_𝑒𝑒𝑀[𝐮𝐯]𝑐][𝑑𝑠,𝐮,𝑝∶=𝑐₀𝐮,𝑝−_Γ𝑝𝑐̃𝑐̃𝛾𝐮𝑑𝑠,𝐮,𝑝∶=₀𝐮,𝑝−_Γ𝑝𝛾𝐮𝑑𝑠, (18) where 𝛼>0 is a parameter to be determined later. For the exact solution (𝐰,𝑞) of system (2), we have 𝑎₀(𝐰,𝐮)=𝑎(𝐰,𝐮),∀𝐮∈𝑈_ℎ,𝑐₀𝐰,𝑝=𝑐𝐰,𝑝,∀𝑝∈𝐻_ℎ,𝑐₀̃𝐰,𝑝=𝑐̃𝐰,𝑝,∀𝑝∈𝐻_ℎ. (19) Therefore, it follows from (17) that 𝑎(𝐰,𝐯)+𝑏𝐯,𝑞=0,∀𝐯∈𝑈_ℎ,𝑞_𝑡,𝑝+𝑐𝐰,𝑝+^𝑡₀𝑐̃𝐰,𝑝𝑑𝜏=𝑓,𝑝,∀𝑝∈𝐻_ℎ. (20)

The discontinuous mixed covolume scheme for (2) reads as follows. Seek (𝐰_ℎ,𝑞_ℎ)∈𝑈_ℎ×𝐻_ℎ such that 𝑎𝐰_ℎ,𝐯+𝑏𝐯,𝑞_ℎ=0,∀𝐯∈𝑈_ℎ,𝑞_ℎ𝑡,𝑝+𝑐𝐰_ℎ,𝑝+^𝑡₀𝑐̃𝐰_ℎ,𝑝𝑑𝜏=𝑓,𝑝,∀𝑝∈𝐻_ℎ, (21) where 𝑞_ℎ(0)=𝑞_ℎ(0), 𝐰_ℎ(0)=𝐰_ℎ(0), 𝑞_ℎ(0),𝐰_ℎ(0) will be given in Section 4.

Let 𝑁>0 be a positive integer; let 0=𝑡⁰<𝑡¹<⋯<𝑡^𝑗<⋯<𝑡^𝑁=𝑇 be a subdivision of time. 𝑡^𝑗=𝑗Δ𝑡(0≤𝑗≤𝑁),Δ𝑡=𝑇/𝑁. We use the backward Euler difference quotient 𝜕_𝑡𝑞^𝑗_ℎ=𝑞^𝑗_ℎ−𝑞_ℎ^𝑗−1Δ𝑡𝑗=1,2,…,𝑁 (22) to approximate the differential quotient 𝜕𝑞^𝑗_ℎ/𝜕𝑡(𝑗=1,2,…,𝑁) and the numerical integration Δ𝑡∑^𝑗−1_𝑘=0∇⋅(𝐷^𝑘𝐰^𝑘_ℎ) to approximate the integration ∫^𝑡𝑗₀∇⋅(𝐷𝐰_ℎ)𝑑𝜏(𝑗=1,2,…,𝑁) in the semidiscrete scheme; then we obtain the backward Euler fully discrete discontinuous mixed covolume scheme for the problem (1): find (𝐰^𝑗_ℎ,𝑞^𝑗_ℎ)∈𝑈_ℎ×𝐻_ℎ(𝑗=1,2,…,𝑁), such that 𝑎𝐰^𝑗_ℎ,𝐯+𝑏𝐯,𝑞^𝑗_ℎ=0,∀𝐯∈𝑈_ℎ,𝜕_𝑡𝑞^𝑗_ℎ,𝑝+𝑐𝐰^𝑗_ℎ,𝑝+𝑐̃Δ𝑡^𝑗−1_𝑘=0𝐰^𝑘_ℎ,𝑝=𝑓,𝑝,∀𝑝∈𝐻_ℎ,𝑗=1,2,…,𝑁, (23) where 𝑞⁰_ℎ=𝑞_ℎ(0), 𝐰⁰_ℎ=𝐰_ℎ(0), 𝑞_ℎ(0), 𝐰_ℎ(0) will be given in Section 4.

We define the following norms for 𝐮∈𝑈(ℎ): |‖𝐮‖|²_div=‖𝐮‖²+∇_ℎ⋅𝐮²+_𝑒∈Γ1ℎ_𝑒𝑒[𝐮]²𝐮𝑑𝑠,|‖‖|²₁=‖𝐮‖²+|𝐮|²_1,ℎ+_𝑒∈Γ1ℎ_𝑒𝑒[𝐮]²𝐮𝑑𝑠,|‖‖|²=|‖𝐮‖|²₁+_{𝐾∈𝒯ℎ}ℎ²_𝐾|𝐮|²_2,𝐾, (24) where ∇_ℎ⋅𝐮 is the function whose restriction to each element 𝐾∈𝒯_ℎ is equal to ∇⋅𝐮 and |𝐮|²_1,ℎ=∑_{𝐾∈𝒯ℎ}|𝐮|²_1,𝐾.

We will introduce some useful Lemmas; for more details, see [6].

Lemma 1. For 𝐮,𝐯∈𝑈(ℎ), we have 𝑎(𝐮,𝐯)≤𝐶|‖𝐮‖|_div|‖𝐯‖|_div. (25)

Lemma 2. For (𝐮,𝑝)∈𝑈(ℎ)×𝐿²(Ω), we have 𝑏𝐮,𝑝=−𝑐𝐮,𝑝. (26)

Lemma 3. For (𝐮,𝑝)∈𝑈(ℎ)×𝐿²(Ω), we have 𝑏𝐮,𝑝≤𝐶|‖𝐮‖|𝑝+_{𝐾∈𝒯ℎ}ℎ²_𝐾𝑝²_1,𝐾^1/2; (27) if (𝐮,𝑝)∈𝑈_ℎ×𝐻_ℎ, then 𝑏𝐮⋅𝑝,𝑐̃𝐮⋅𝑝.𝐮,𝑝≤𝐶|‖‖|𝐮,𝑝≤𝐶|‖‖| (28)

Lemma 4. Let 𝑍_ℎ={𝐮∣𝐮∈𝑈_ℎ,𝑐(𝐮,𝑝)=0,∀𝑝∈𝐻_ℎ}; for any 𝐮∈𝑍_ℎ, there is a constant 𝐶₀ independent of ℎ such that, for 𝛼 is large enough, 𝑎(𝐮,𝐮)≥𝐶₀|‖𝐮‖|²_div. (29)

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 𝑈_ℎ×𝐻_ℎ.

Theorem 5. The semidiscrete discontinuous mixed covolume scheme (21) has a unique solution in the space 𝑈_ℎ×𝐻_ℎ.

Proof. Only prove that homogenous equation 𝑎𝐰_ℎ,𝐮+𝑏𝐮,𝑞_ℎ=0,∀𝐮∈𝑈_ℎ,𝑞_ℎ𝑡,𝑝+𝑐𝐰_ℎ,𝑝+^𝑡₀𝑐₀̃𝐰_ℎ,𝑝=0,∀𝑝∈𝐻_ℎ,𝑞_ℎ(0)=0,𝐰_ℎ(0)=0, (30) of (21) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting 𝐯=𝐰_ℎ in the first formula of (30) and 𝑝=𝑞_ℎ in the second formula of (30), using Lemma 2, the sum of (30) gives 𝑎𝐰_ℎ,𝐰_ℎ+𝑞_ℎ𝑡,𝑞_ℎ=−^𝑡₀𝑐₀̃𝐰_ℎ,𝑞_ℎ. (31) Using (1/2)(𝑑/𝑑𝑡)(𝑞_ℎ,𝑞_ℎ)=(𝑞_ℎ𝑡,𝑞_ℎ) and Lemmas 3 and 4, we have that 𝐶₀𝐰_ℎ²_div+12𝑑𝑞𝑑𝑡^ℎ2≤𝐶^𝑡₀𝐰_ℎ𝑞_ℎ𝑑𝜏≤𝐶^𝑡₀𝐰_ℎdiv𝑞_ℎ𝑑𝜏. (32) Using Hölder inequality and Gronwall Lemma, we get 𝐶₀𝐰_ℎ²_div+12𝑑𝑞𝑑𝑡^ℎ2≤𝐶^𝑡₀𝑞_ℎ²𝑑𝜏. (33) Integrating the above formula, we get 2𝐶₀^𝑡₀𝐰_ℎ²_div𝑑𝜏+𝑞_ℎ²≤𝐶^𝑡₀^𝑠₀𝑞_ℎ²𝑑𝜏𝑑𝑠≤𝐶^𝑡₀𝑞_ℎ²𝑑𝜏. (34) Then ‖𝑞_ℎ‖=0, |‖𝐰_ℎ‖|_div=0. So 𝑞_ℎ=0, 𝐰_ℎ=𝟎, 𝑡∈(0,𝑇]. 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 𝑈_ℎ×𝐻_ℎ if Δ𝑡 is sufficiently small.

Proof. Only prove that homogenous equation 𝑎𝐰^𝑗_ℎ,𝐯+𝑏𝐯,𝑞^𝑗_ℎ=0,∀𝐯∈𝑈_ℎ,𝜕_𝑡𝑞^𝑗_ℎ,𝑝+𝑐𝐰^𝑗_ℎ,𝑝+𝑐̃Δ𝑡^𝑗−1_𝑘=0𝐰^𝑘_ℎ,𝑝=0,∀𝑝∈𝐻_ℎ,𝑗=1,2,…,𝑁, (35) of (23) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting 𝐯=𝐰^𝑗_ℎ in the first formula of (35) and 𝑝=𝑞^𝑗_ℎ in the second formula of (35), using Lemma 2, the sum of (35) gives 𝑎𝐰^𝑗_ℎ,𝐰^𝑗_ℎ+𝜕_𝑡𝑞^𝑗_ℎ,𝑞^𝑗_ℎ=−𝑐̃Δ𝑡^𝑗−1_𝑘=0𝐰^𝑘_ℎ,𝑞^𝑗_ℎ,𝑗=1,2,…,𝑁. (36) Using Lemmas 3 and 4 and 𝜕_𝑡𝑞^𝑗_ℎ,𝑞^𝑗_ℎ=1𝑞Δ𝑡^𝑗ℎ−𝑞_ℎ^𝑗−1,𝑞^𝑗_ℎ=1𝑞2Δ𝑡^𝑗ℎ,𝑞^𝑗_ℎ−𝑞_ℎ^𝑗−1,𝑞_ℎ^𝑗−1+𝑞^𝑗_ℎ−𝑞_ℎ^𝑗−1,𝑞^𝑗_ℎ−𝑞_ℎ^𝑗−1=1𝑞2Δ𝑡^𝑗ℎ2−𝑞_ℎ^𝑗−12+𝑞^𝑗_ℎ−𝑞_ℎ^𝑗−12≥1𝑞2Δ𝑡^𝑗ℎ2−𝑞_ℎ^𝑗−12, (37) we have from (36) that 2𝐶₀Δ𝑡𝐰^𝑗_ℎ²_div+𝑞^𝑗_ℎ²−𝑞_ℎ^𝑗−12≤𝐶Δ𝑡(Δ𝑡)^2𝑗−1_𝑘=0𝐰^𝑘_ℎ²_div+𝑞^𝑗_ℎ²,𝑗=1,2,…,𝑁. (38) Adding the above inequality with 𝑗 from 1 to 𝑖, using 𝑞⁰_ℎ=0 and the discrete Gronwall inequality, when Δ𝑡 is sufficiently small, we have 2𝐶₀Δ𝑡^𝑖_𝑗=1𝐰^𝑗_ℎ²_div+𝑞^𝑖_ℎ²≤0(𝑖=1,2,…,𝑁). (39) Hence we have ‖𝑞^𝑖_ℎ‖²=0 and |‖𝐰^𝑖_ℎ‖|²_div=0(𝑖=1,2,…,𝑁); that is, 𝑞^𝑖_ℎ=0 and 𝐰^𝑖_ℎ=𝟎(𝑖=1,2,…,𝑁). This completes the proof.

4. A Discontinuous Mixed Covolume Elliptic Projection

Define an operator 𝜋_𝐾 from 𝐻¹(𝐾) to 𝑃₁(𝐾) by requiring that, for any ∀𝑢∈𝐻¹(𝐾), _𝑒𝑖𝜋_𝐾𝑢𝑑𝑠=_𝑒𝑖𝑢𝑑𝑠,(𝑖=1,2,3), (40) where 𝑒_𝑖(𝑖=1,2,3) are the three sides of the element 𝐾∈𝒯_ℎ. It was proved in [5] that 𝜋_𝐾𝑢−𝑢_𝑠,𝐾≤ℎ^2−𝑠|𝑢|_2,𝐾,∀𝑢∈𝐻²(𝐾),(𝑠=0,1,2). (41)

For any 𝐮∈𝐻¹₀(Ω)², define Π₁𝐮∈𝑈_ℎ by Π₁𝐮_𝑖𝐾=Π_𝐾𝑢_𝑖,∀𝐾∈𝒯_ℎ,(𝑖=1,2). (42) Using the definition of Π₁ and integration by parts, we can show that 𝑐𝐮−Π₁𝐮,𝑝=0,∀𝑝∈𝐻_ℎ. (43) It was proved in [6] that 𝐰−Π₁𝐰_div≤𝑐ℎ‖𝐰‖₂. (44) Let Π₂ be the projection from 𝐿²₀(Ω) to the finite element space 𝐻_ℎ.

Define a discontinuous mixed covolume elliptic projection by requiring that, finding 𝐰_ℎ,𝑞_𝑛: (0,𝑡)→𝑈_ℎ×𝐻_ℎ, such that 𝑎𝐰−𝐰_ℎ,𝐯+𝑏𝐯,𝑞−𝑞_ℎ=0,∀𝐯∈𝑈_ℎ,𝑐𝐰−𝐰_ℎ,𝑝=0,∀𝑝∈𝐻_ℎ. (45) It was proved in [15] that the above formula has a unique solution and the error estimates in the following Theorem 7.

Theorem 7. Let (𝐰_ℎ,𝑞_ℎ)∈𝑈_ℎ×𝐻_ℎ be the solution of (45) and (𝐰,𝑞)∈𝐻²(Ω)²×𝐻¹(Ω) the solution of (20). Then there exists a positive constant 𝐶 independent of ℎ such that 𝐰−𝐰_ℎdiv+𝑞−𝑞_ℎ≤𝐶ℎ‖𝐰‖₂+𝑞₁. (46)

Theorem 8. Let (𝐰_ℎ,𝑞_ℎ)∈𝑈_ℎ×𝐻_ℎ be the solution of (45) and (𝐰,𝑞)∈𝐻²(Ω)²×𝐻¹(Ω) the solution of (20). Then there exists a positive constant 𝐶 independent of ℎ such that (𝐰−𝐰_ℎ)_𝑡div+𝑞−𝑞_ℎ𝑡≤𝐶ℎ𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁. (47)

Differentiating each equation of (45) on 𝑡 and using (43), (44) we can prove this theorem in the same way as [15].

5. Error Estimates for Semidiscrete Method

In this section, we will establish the error estimates in the 𝐻(div) and 𝐿² norms for the semidiscrete discontinuous mixed covolume method.

Theorem 9. Let (𝐰_ℎ,𝑞_ℎ)∈𝑈_ℎ×𝐻_ℎ be the solution of (21) and 𝑞_ℎ(0)=𝑞_ℎ(0), (𝐰,𝑞)∈𝐻²(Ω)²×𝐻¹(Ω) the solution of (2). Then there exists a positive constant 𝐶 independent of ℎ such that 𝐰−𝐰_ℎdiv+𝑞−𝑞_ℎ≤𝐶ℎ^𝑡₀𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝑑𝜏+‖𝐰‖₂+𝑞₁. (48)

Proof. Let 𝜉=𝑞_ℎ−𝑞_ℎ, 𝜂=𝐰_ℎ−𝐰_ℎ. Subtracting the two equations of (21) from those of (20), respectively, we have 𝑎𝐰−𝐰_ℎ,𝐯+𝑏𝐯,𝑞−𝑞_ℎ=0,∀𝐯∈𝑈_ℎ,𝑞_𝑡−𝑞_ℎ𝑡,𝑝+𝑐𝐰−𝐰_ℎ,𝑝+^𝑡₀𝑐̃𝐰−𝐰_ℎ,𝑝𝑑𝜏=0,∀𝑝∈𝐻_ℎ. (49) Using (45), we have 𝑎𝜂,𝐯+𝑏(𝐯,𝜉)=0,∀𝐯∈𝑈_ℎ,𝑞_𝑡−𝑞_ℎ𝑡,𝑝+𝑐𝜂,𝑝=−^𝑡₀𝑐̃𝜂,𝑝𝑑𝜏−^𝑡₀𝑐̃𝐰−𝐰_ℎ,𝑝𝑑𝜏,∀𝑝∈𝐻_ℎ. (50) Differentiating the first equation of (50) on 𝑡, we have that 𝑎𝜂_𝑡,𝐯+𝑏𝐯,𝜉_𝑡=0,∀𝐯∈𝑈_ℎ. (51)
By letting 𝑝=𝜉_𝑡 in the second formula of (50) and letting 𝐯=𝜂 in (51), using Lemma 2, the sum of them gives 𝑎𝜂_𝑡,𝜂+𝜉_𝑡,𝜉_𝑡=𝑝_ℎ−𝑝_𝑡,𝜉_𝑡−^𝑡₀𝑐̃𝜂,𝜉_𝑡𝑑𝜏−^𝑡₀𝑐̃𝐰−𝐰_ℎ,𝜉_𝑡𝑑𝜏. (52) Using 𝑎𝜂_𝑡=1,𝜂2𝑑𝑎−1𝑑𝑡𝜂,𝜂2𝑀_𝑡𝜂,𝜂 (53) and Lemmas 1 and 3 gives 12𝑑𝑎+𝜉𝑑𝑡𝜂,𝜂_𝑡²≤𝐶𝜉_𝑡⋅𝑝_ℎ−𝑝_𝑡+𝜂²_div+^𝑡₀𝜂_div𝜉_𝑡𝑑𝜏+^𝑡₀𝐰−𝐰_ℎdiv𝜉_𝑡𝑑𝜏. (54) Multiplying the equation above with 2, integrating them from 0 to 𝑡 and using Hölder inequality, 𝜖-inequality, Gronwall inequality, Lemma 4, and (47), we can get 𝜂²_div+1𝐶₀^𝑡₀𝜉_𝑡𝑑𝜏²≤𝐶ℎ^2𝑡₀𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝑑𝜏², (55) so 𝜂_div≤𝐶ℎ^𝑡₀𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝑑𝜏,_𝑡0𝜉_𝑡𝑑𝜏≤𝐶ℎ^𝑡₀𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝑑𝜏; (56)(57) hence 𝜉≤^𝑡₀𝜉_𝑡𝑑𝜏≤𝐶ℎ^𝑡₀𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝑑𝜏. (58)
Now, using the triangle inequality, (46), (56), and (58), we get 𝐰−𝐰_ℎdiv+𝑞−𝑞_ℎ≤𝐶ℎ^𝑡₀𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝑑𝜏+‖𝐰‖₂+𝑞₁. (59) The proof is complete.

6. Error Estimates for Fully Discrete Method

Let 𝜉^𝑗=𝑞^𝑗_ℎ−𝑞^𝑗_ℎ, 𝜁^𝑗=𝑞^𝑗_ℎ−𝑞^𝑗, 𝜂^𝑗=𝐰^𝑗_ℎ−𝐰^𝑗_ℎ(𝑗=0,1,…,𝑁), and then the error estimates for the backward Euler fully discrete discontinuous mixed covolume method in the 𝐻(div) and 𝐿² norms are provided in next two theorems.

Theorem 10. Let (𝐰,𝑞)∈𝐻²(Ω)²×𝐻¹(Ω) be the solution of (2) and (𝐰^𝑗_ℎ,𝑞^𝑗_ℎ)∈𝑈_ℎ×𝐻_ℎ(𝑗=1,2,…,𝑁) the solution of (23) with 𝑡=𝑡_𝑗(𝑗=1,2,…,𝑁), respectively. If 𝑞⁰_ℎ=𝑞_ℎ(0)=𝑞₀, 𝐰⁰_ℎ=𝐰_ℎ(0)=𝐰₀, then there exists a positive constant 𝐶 independent of ℎ and Δ𝑡 such that max_{0≤𝑖≤𝑁}𝑞^𝑖−𝑞^𝑖_ℎ+max_{0≤𝑖≤𝑁}𝐰^𝑖−𝐰^𝑖_ℎdiv≤𝐶Δ𝑡𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝐰_{𝑡𝐿∞(𝐻2)}+𝐶ℎ𝐰_{𝑡𝐿∞(𝐻2)}+𝑞_{𝑡𝐿∞(𝐻1)}+‖𝐰‖_𝐿^∞_(𝐻²₎+𝑞_{𝐿∞(𝐻1)}. (60)

Proof. Subtracting the two equations of (23) from (20), respectively, with 𝑡=𝑡^𝑗(𝑗=0,1,…,𝑁), we can get the error equation: 𝑎𝐰^𝑗−𝐰^𝑗_ℎ,𝐯+𝑏𝐯,𝑞^𝑗−𝑞^𝑗_ℎ=0,∀𝐯∈𝑈_ℎ,𝑞^𝑗_𝑡−𝜕_𝑡𝑞^𝑗_ℎ,𝑝+𝑐𝐰^𝑗−𝐰^𝑗_ℎ,𝑝+^𝑡𝑗₀𝑐̃−𝑐̃𝐰,𝑝𝑑𝜏Δ𝑡^𝑗−1_𝑘=0𝐰^𝑘_ℎ,𝑝=0,∀𝑝∈𝐻_ℎ,𝑗=1,2,…,𝑁. (61)
Choosing 𝐯=𝜂^𝑗(𝑗=1,2,…,𝑁) and 𝑝=𝜉^𝑗(𝑗=1,2,…,𝑁) in the two equations of (61), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection with 𝑡=𝑡^𝑗(𝑗=0,1,…,𝑁), we have 𝑎𝜂^𝑗,𝜂^𝑗+𝜕_𝑡𝜉^𝑗,𝜉^𝑗=𝜕_𝑡𝑞^𝑗−𝑞^𝑗_𝑡,𝜉^𝑗+𝜕_𝑡𝜁^𝑗,𝜉^𝑗−^𝑡𝑗₀𝑐̃𝐰−𝐰_ℎ,𝜉^𝑗−𝑐̃𝑑𝜏Δ𝑡^𝑗−1_𝑘=0𝜂^𝑘,𝜉^𝑗−^𝑡𝑗₀𝑐̃𝐰_ℎ,𝜉^𝑗𝑑𝜏+𝑐̃Δ𝑡^𝑗−1_𝑘=0𝐰^𝑘_ℎ,𝜉^𝑗. (62)
First, we estimate the left item of (62). Using Lemma 4, we have 𝑎𝜂^𝑗,𝜂^𝑗≥𝐶₀𝜂^𝑗2_div,𝜕_𝑡𝜉^𝑗,𝜉^𝑗=1𝜉Δ𝑡^𝑗−𝜉^𝑗−1,𝜉^𝑗=1𝜉2Δ𝑡^𝑗,𝜉^𝑗−𝜉^𝑗−1,𝜉^𝑗−1+𝜉^𝑗−𝜉^𝑗−1,𝜉^𝑗−𝜉^𝑗−1=1𝜉2Δ𝑡^𝑗2−𝜉^𝑗−12+𝜉^𝑗−𝜉^𝑗−12>1𝜉2Δ𝑡^𝑗2−𝜉^𝑗−12. (63)
Then, we estimate the right item of (62). From 𝜕_𝑡𝑞^𝑗−𝑞^𝑗_𝑡²=1Δ𝑡^𝑡𝑗_𝑡^𝑗−1(𝑡^𝑗−1−𝑡)𝑞_𝑡𝑡𝑑𝑡²≤_Ω1Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝑡^𝑗−1−𝑡𝑞_𝑡𝑡𝑑𝑡²≤1𝑑𝑥(Δ𝑡)^{2Ω𝑡𝑗𝑡𝑗−1}𝑡^𝑗−1−𝑡²𝑑𝑡^𝑡𝑗_𝑡𝑗−1𝑞²_𝑡𝑡()𝑑𝑡𝑑𝑥≤𝐶Δ𝑡²𝑞_𝑡𝑡²_{𝐿∞(𝐿2)}, (64) we have 𝜕_𝑡𝑞^𝑗−𝑞^𝑗_𝑡,𝜉^𝑗≤𝐶(Δ𝑡)𝑞_{𝑡𝑡𝐿∞(𝐿2)}𝜉^𝑗,𝜕_𝑡𝜁^𝑗=𝜁^𝑗−𝜁^𝑗−1=1Δ𝑡Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝜁_𝑡≤1𝑑𝑡Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝜁_𝑡≤𝑑𝑡𝐶ℎΔ𝑡^𝑡𝑗_𝑡^𝑗−1𝐰_𝑡2+𝑞_𝑡1+‖𝐰‖₂+𝑞₁𝐰𝑑𝑡≤𝐶ℎ_{𝑡𝐿∞(𝐻2)}+𝑞_{𝑡𝐿∞(𝐻1)}+‖𝐰‖_𝐿^∞_(𝐻²₎+𝑞_{𝐿∞(𝐻1)}, (65) and therefore 𝜕_𝑡𝜁^𝑗,𝜉^𝑗≤𝐶ℎ𝐰_{𝑡𝐿∞(𝐻2)}+𝑞_{𝑡𝐿∞(𝐻1)}+‖𝐰‖_𝐿^∞_(𝐻²₎+𝑞_{𝐿∞(𝐻1)}×𝜉^𝑗. (66) Using Lemma 3, we can get −^𝑡𝑗₀𝑐̃𝐰−𝐰_ℎ,𝜉^𝑗𝑑𝜏≤𝐶^𝑡𝑗₀𝐰−𝐰_ℎdiv𝑑𝜏𝜉^𝑗≤𝐶ℎ‖𝐰‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}𝜉^𝑗,−𝑐̃Δ𝑡^𝑗−1_𝑘=0𝜂^𝑘,𝜉^𝑗≤𝐶(Δ𝑡)^𝑗−1_𝑘=0𝜂^𝑘_div𝜉^𝑗,−^𝑡𝑗₀𝑐̃𝐰_ℎ,𝜉^𝑗𝑑𝜏+𝑐̃Δ𝑡^𝑗−1_𝑘=0𝐰^𝑘_ℎ,𝜉^𝑗=𝑐̃^𝑗−1_𝑘=0^𝑡𝑘+1_𝑡𝑘𝐰^𝑘_ℎ−𝐰_ℎ(𝜏)𝑑𝜏,𝜉^𝑗≤𝐶^𝑗−1_𝑘=0^𝑡𝑘+1_𝑡𝑘^𝜏_𝑡𝑘𝐰_ℎ𝑡𝑑𝑠𝑑𝜏𝜉^𝑗≤𝐶(Δ𝑡)𝐰_{𝑡𝐿∞(𝐻2)}𝜉^𝑗. (6)
Substituting the estimations above into (62), multiplying them with Δ𝑡, adding them with 𝑗 from 1 to 𝑖, and using 𝜉⁰=0, we have 𝜉^𝑖2+2𝐶₀Δ𝑡^𝑖_𝑗=1𝜂^𝑗2_div≤𝐶(Δ𝑡)²𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝐰_{𝑡𝐿∞(𝐻2)}²+𝐶(Δ𝑡)^2𝑗−1_𝑘=0𝜂^𝑘2_div+𝐶Δ𝑡^𝑖_𝑗=1𝜉^𝑗2+𝐶ℎ²𝐰_{𝑡𝐿∞(𝐻2)}+𝑞_{𝑡𝐿∞(𝐻1)}+‖𝐰‖_𝐿^∞_(𝐻²₎+𝑞_{𝐿∞(𝐻1)}². (68) Using the discrete Gronwall inequality, we have max_{0≤𝑖≤𝑁}𝜉^𝑖+max_{0≤𝑖≤𝑁}𝜂^𝑖_div≤𝐶Δ𝑡𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝐰_{𝑡𝐿∞(𝐻2)}+𝐶ℎ𝐰_{𝑡𝐿∞(𝐻2)}+𝑞_{𝑡𝐿∞(𝐻1)}+‖𝐰‖_𝐿^∞_(𝐻²₎+𝑞_{𝐿∞(𝐻1)}. (69) From the formula above and (46) and using the triangle inequality, we have max_{0≤𝑖≤𝑁}𝑞^𝑖−𝑞^𝑖_ℎ+max_{0≤𝑖≤𝑁}𝐰^𝑖−𝐰^𝑖_ℎdiv≤𝐶Δ𝑡𝑞_{𝑡𝑡𝐿∞(𝐿2)}+𝐰_{𝑡𝐿∞(𝐻2)}+𝐶ℎ𝐰_{𝑡𝐿∞(𝐻2)}+𝑞_{𝑡𝐿∞(𝐻1)}+‖𝐰‖_𝐿^∞_(𝐻²₎+𝑞_{𝐿∞(𝐻1)}. (70) This completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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).