Discontinuous Mixed Covolume Methods for Parabolic Problems

Abstract

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 𝐿².

1. 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 [2–10]. 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 𝐻(div) and first-order 𝐿²-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 𝐻(div) and 𝐿² norms of semidiscrete method and fully discrete method are proved in Sections 5 and 6.

Throughout this paper, 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

In this paper, we consider the following parabolic problems: 𝑝_𝑡()𝐷()((),()(],𝑝()𝐱,𝑡−∇⋅𝐱,𝑡∇𝑝𝐱,𝑡)=𝑓𝐱,𝑡𝐱,𝑡∈Ω×0,𝑇𝐱,0=𝑝₀(𝐱)𝑝()()(],,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),  𝑝 is an unknown function, and 𝐷 is a symmetric, bounded matrix function which satisfies the following condition: there exist two positive constants 𝛼₁, 𝛼₂ such that 𝛼₁𝜉^⊤𝜉≤𝜉^⊤𝐷𝜉≤𝛼₂𝜉^⊤𝜉,∀𝜉=𝜉₁,𝜉₂∈𝐑²; (2)𝑓 is a given function in 𝐿²(Ω). Furthermore, we assume that the matrix 𝑀=𝐷⁻¹ is locally Lipschitz.

Here and in what follows, we will not write the independent 𝐱, 𝑡 for any functions unless it is necessary.

Let 𝐮=−𝐷∇𝑝, and rewrite (1) as the system of first-order partial differential equations: ()(],𝑝𝑀𝐮+∇𝑝=0,𝐱,𝑡∈Ω×0,𝑇_𝑡()(],𝑝()+∇⋅𝐮=𝑓,𝐱,𝑡∈Ω×0,𝑇𝐱,0=𝑝₀(𝐱)𝑝()()(].,𝐱∈Ω,𝐱,𝑡=0,𝐱,𝑡∈𝜕Ω×0,𝑇 (3)

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. 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. 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 𝑉_ℎ∶=𝐯∈𝐿²(Ω)²∶𝐯|_𝐾∈𝑃₁(𝐾)²,∀𝐾∈𝒯_ℎ. (4) Define the finite-dimensional test function space 𝑊_ℎ for velocity associated with the dual partition 𝒯^∗_ℎ as 𝑊_ℎ∶=𝐰∈𝐿²(Ω)²∶𝐰|_𝑇∈𝑃₀(𝑇)²,∀𝑇∈𝒯^∗_ℎ. (5) Let 𝐻_ℎ be the finite-dimensional space for pressure: 𝐻_ℎ∶=𝑞∈𝐿²(Ω)∶𝑞_𝐾∈𝑃₀(𝐾),∀𝐾∈𝒯_ℎ. (6)

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, 𝑞=12𝑞_𝜕𝐾1+𝑞_𝜕𝐾2,𝑞=𝑞_𝜕𝐾1𝐧₁+𝑞_𝜕𝐾2𝐧₂,{𝐯}=12𝐯|_𝜕𝐾1+𝐯|_𝜕𝐾2,[𝐯]=𝐯|_𝜕𝐾1𝐧₁+𝐯|_𝜕𝐾2𝐧₂. (7) If 𝑒 is an edge on the boundary of Ω, we set 𝑞=𝑞,[𝐯]=𝐯⋅𝐧, (8) where 𝐧 is the outward unit normal. We do not require either of the quantities [𝑞] or {𝐯} on boundary edges, and we leave them undefined.

Multiplying the first and second equations in system (3) by 𝐰∈𝑊_ℎ and 𝑞∈𝐻_ℎ, respectively, and using the integration by parts formula in the equation, we have _{𝑇∈𝒯∗ℎ𝑇}𝑀𝐮⋅𝐰𝑑𝑥+_{𝑇∈𝒯∗ℎ𝜕𝑇}𝑝𝐰⋅𝐧𝑑𝑠=0,∀𝐰∈𝑊_ℎ,_{𝐾∈𝒯ℎ𝐾}𝑝_𝑡𝑞𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}∇⋅𝐮𝑞𝑑𝑥=𝑓,𝑞,∀𝑞∈𝐻_ℎ, (9) 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}𝑝𝐰⋅𝐧𝑑𝑠=_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}+𝑝𝐰⋅𝐧𝑑𝑠_{𝐾∈𝒯ℎ𝜕𝐾}𝑝𝐰⋅𝐧𝑑𝑠,∀𝐰∈𝑊_ℎ, (10) where 𝐴₄=𝐴₁. A straightforward computation gives _{𝐾∈𝒯ℎ𝜕𝐾}𝑝𝐰⋅𝐧𝑑𝑠=_{𝑒∈Γ0𝑒}𝑝⋅{𝐰}𝑑𝑠+_{𝑒∈Γ𝑒}𝑝⋅[𝐰]𝑑𝑠,∀𝐰∈𝑊_ℎ. (11) Let ∫_Γ𝑝𝑑𝑠=∑_𝑒∈Γ∫_𝑒𝑝𝑑𝑠. Using the above formula and the fact that [𝑝]=0 for 𝑝∈𝐻¹(Ω) on Γ⁰, (10) becomes _{𝑇∈𝒯∗ℎ𝜕𝑇}𝑝𝐰⋅𝐧𝑑𝑠=_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}+𝑝𝐰⋅𝐧𝑑𝑠_{𝑒∈Γ𝑒}𝑝[𝐰]𝑑𝑠,∀𝐰∈𝑊_ℎ. (12) Then, system (9) can be rewritten as follows: _{𝑇∈𝒯∗ℎ𝑇}𝑀𝐮⋅𝐰𝑑𝑥+_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}+𝑝𝐰⋅𝐧𝑑𝑠_{𝑒∈Γ𝑒}𝑝[𝐰]𝑑𝑠=0,∀𝐰∈𝑊_ℎ,_{𝐾∈𝒯ℎ𝐾}𝑝_𝑡𝑞𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}∇⋅𝐮𝑞𝑑𝑥=𝑓,𝑞,∀𝑞∈𝐻_ℎ. (13)

Let 𝑉(ℎ)=𝑉_ℎ+𝐻²(Ω)². Define a mapping 𝛾∶𝑉(ℎ)→𝑊_ℎ as 𝛾𝐯_𝑇=1ℎ_𝑒𝑒𝐯|_𝑇𝑑𝑠,𝑇∈𝒯^∗_ℎ, (14) where ℎ_𝑒 is the length of the edge 𝑒. For 𝐯=(𝑣₁,𝑣₂)∈𝑉(ℎ), 𝛾𝑣_𝑖(𝑖=1,2) is defined as 𝛾𝑣_𝑖𝑇=1ℎ_𝑒𝑒𝑣_𝑖𝑇𝑑𝑠,𝑇∈𝒯^∗_ℎ,(𝑖=1,2). (15) Then the system (13) is equivalent to _{𝑇∈𝒯∗ℎ𝑇}𝑀𝐮⋅𝛾𝐰𝑑𝑥+_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}+𝑝𝛾𝐰⋅𝐧𝑑𝑠_{𝑒∈Γ𝑒}𝑝𝛾𝐰𝑑𝑠=0,∀𝐰∈𝑉_{𝐾∈𝒯ℎ𝐾}(ℎ),𝑝_𝑡𝑞𝑑𝑥+_{𝐾∈𝒯ℎ𝐾}∇⋅𝐮𝑞𝑑𝑥=𝑓,𝑞,∀𝑞∈𝐻_ℎ. (16) Let 𝑎₀(𝐯,𝐰)∶=_{𝑇∈𝒯∗ℎ𝑇}𝑏𝑀𝐯⋅𝛾𝐰𝑑𝑥,𝐰,𝑞∶=_{𝐾∈𝒯ℎ}³_{𝑗=1𝐴𝑗+1𝑄𝐴𝑗}𝑞𝛾𝐰⋅𝐧𝑑𝑠+_{𝑒∈Γ𝑒}𝑞𝑐𝛾𝐰𝑑𝑠,₀𝐯,𝑞∶=_{𝐾∈𝒯ℎ𝐾}∇⋅𝐯𝑞𝑑𝑥. (17)

Using the above bilinear forms, it is clear that system (16) can be rewritten as follows: 𝑎₀()(ℎ),𝑝𝐮,𝐰+𝑏𝐰,𝑝=0,∀𝐰∈𝑉_𝑡,𝑞+𝑐₀𝐮,𝑞=𝑓,𝑞,∀𝑞∈𝐻_ℎ. (18)

In order to define our numerical schemes, we introduce the bilinear forms as follows: 𝑎(𝐯,𝐰)∶=𝑎₀(𝐯,𝐰)+𝛼_𝑒∈Γ1ℎ_𝑒𝑒𝑀[𝐯𝐰]𝑐][𝑑𝑠,𝐯,𝑞∶=𝑐₀𝐯,𝑞−_Γ𝑞𝛾𝐯𝑑𝑠, (19) where 𝛼>0 is a parameter to be determined later. For the exact solution (𝐮,𝑝) of system (3), we have 𝑎₀(𝐮,𝐯)=𝑎(𝐮,𝐯),∀𝐯∈𝑉_ℎ,𝑐₀𝐮,𝑞=𝑐𝐮,𝑞,∀𝑞∈𝐻_ℎ. (20) Therefore, it follows from (18) that 𝑎(𝐮,𝐰)+𝑏𝐰,𝑝=0,∀𝐰∈𝑉_ℎ,𝑝_𝑡,𝑞+𝑐𝐮,𝑞=𝑓,𝑞,∀𝑞∈𝐻_ℎ. (21)

The discontinuous mixed covolume scheme for (3) reads as follows. Seek (𝐮_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝐻_ℎ such that 𝑎𝐮_ℎ,𝐰+𝑏𝐰,𝑝_ℎ=0,∀𝐰∈𝑉_ℎ,𝑝_ℎ𝑡,𝑞+𝑐𝐮_ℎ,𝑞=𝑓,𝑞,∀𝑞∈𝐻_ℎ, (22) where 𝑝_ℎ(0)=𝑝_ℎ(0), 𝐮_ℎ(0)=𝐮_ℎ(0), 𝑝_ℎ(0), 𝐮_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 𝜕_𝑡𝑝^𝑗_ℎ=𝑝^𝑗_ℎ−𝑝_ℎ^𝑗−1,Δ𝑡𝑗=1,2,…,𝑁 (23) to approximate the differential quotient (𝜕𝑝^𝑗_ℎ/𝜕𝑡)(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 (𝐮^𝑗_ℎ,𝑝^𝑗_ℎ)∈𝑉_ℎ×𝐻_ℎ,  (𝑗=1,2,…,𝑁), such that 𝑎𝐮^𝑗_ℎ,𝐰+𝑏𝐰,𝑝^𝑗_ℎ=0,∀𝐰∈𝑉_ℎ,𝜕_𝑡𝑝^𝑗_ℎ,𝑞+𝑐𝐮^𝑗_ℎ,𝑞=𝑓,𝑞,∀𝑞∈𝐻_ℎ,𝑗=1,2,…,𝑁, (24) where 𝑝⁰_ℎ=𝑝_ℎ(0), 𝐮⁰_ℎ=𝐮_ℎ(0),𝑝_ℎ(0),𝐮_ℎ(0) will be given in Section 4.

We define the following norms for 𝐯∈𝑉(ℎ): |||𝐯|||²_div=‖𝐯‖²+∇_ℎ⋅𝐯²+_𝑒∈Γ1ℎ_𝑒𝑒[𝐯]²𝐯𝑑𝑠,||||||²₁=‖𝐯‖²+|𝐯|²_1,ℎ+_𝑒∈Γ1ℎ_𝑒𝑒[𝐯]²𝐯𝑑𝑠,||||||²=|||𝐯|||²₁+_{𝐾∈𝒯ℎ}ℎ²_𝐾|𝐯|²_2,𝐾, (25) where ∇_ℎ⋅𝐯 is the function whose restriction to each element 𝐾∈𝒯_ℎ is equal to ∇⋅𝐯, and |𝐯|²_1,ℎ=∑_{𝐾∈𝒯ℎ}|𝐯|²_1,𝐾.

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

Lemma 1. For 𝐯,𝐰∈𝑉(ℎ), one has 𝑎(𝐯,𝐰)≤𝐶|||𝐯|||_div|||𝐰|||_div. (26)

Lemma 2. For (𝐯,𝑞)∈𝑉(ℎ)×𝐿²(Ω), one has 𝑏𝐯,𝑞=−𝑐𝐯,𝑞. (27)

Lemma 3. For (𝐯,𝑞)∈𝑉(ℎ)×𝐿²(Ω), one has 𝑏𝐯,𝑞≤𝐶|||𝐯|||𝑞+_{𝐾∈𝒯ℎ}ℎ²_𝐾𝑞²_1,𝐾^1/2; (28) if (𝐯,𝑞)∈𝑉_ℎ×𝐻_ℎ, then 𝑏𝐯,𝑞≤𝐶|||𝐯|||⋅𝑞. (29)

Lemma 4. Let 𝑍_ℎ={𝐯𝐯∈𝑉_ℎ,𝑐(𝐯,𝑞)=0,∀𝑞∈𝐻_ℎ}; for any 𝐯∈𝑍_ℎ, there is a constant 𝐶₀ independent of ℎ such that, for 𝛼 is large enough, 𝑎(𝐯,𝐯)≥𝐶₀|||𝐯|||²_div. (30)

Lemma 5. For any 𝑞∈𝐻_ℎ, there is a constant 𝛽₀ independent of ℎ such that sup_{𝐯∈𝑉ℎ}𝑐𝐯,𝑞|||𝐯|||≥𝛽₀𝑞. (31)

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 𝑉_ℎ×𝐻_ℎ.

Theorem 6. Semidiscrete discontinuous mixed covolume scheme (22) has a unique solution in the space 𝑉_ℎ×𝐻_ℎ.

Proof. Only prove that homogenous equation 𝑎𝐮_ℎ,𝐰+𝑏𝐰,𝑝_ℎ=0,∀𝐰∈𝑉_ℎ,𝑝_ℎ𝑡,𝑞+𝑐𝐮_ℎ,𝑞=0,∀𝑞∈𝐻_ℎ,𝑝_ℎ(0)=0,𝐮_ℎ(0)=0 (32) of (22) exists unique zero solution since the number of unknowns is the same as the number of line equations.
By letting 𝐰=𝐮_ℎ in the first formula of (32) and 𝑞=𝑝_ℎ in the second formula of (32), using Lemma 2, the sum of (32) gives 𝑝_ℎ𝑡,𝑝_ℎ+𝑎𝐮_ℎ,𝐮_ℎ=0. (33) Using (1/2)(𝑑/𝑑𝑡)(𝑝_ℎ,𝑝_ℎ)=(𝑝_ℎ𝑡,𝑝_ℎ) and Lemma 4, we have 12𝑑𝑝𝑑𝑡^ℎ2+𝐶₀𝐮_ℎ²_div≤0. (34) Integrating the above formula, we get 𝑝_ℎ²+2𝐶₀^𝑡₀𝐮_ℎ²_div𝑑𝜏≤0. (35) Then ‖𝑝_ℎ‖=0, |||𝐮_ℎ|||_div=0. So, 𝑝_ℎ=0, 𝐮_ℎ=𝟎, 𝑡∈(0,𝑇]. 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 𝑉_ℎ×𝐻_ℎ.

Proof. Only prove that homogenous equation 𝑎𝐮^𝑗_ℎ,𝐰+𝑏𝐰,𝑝^𝑗_ℎ=0,∀𝐰∈𝑉_ℎ,𝜕_𝑡𝑝^𝑗_ℎ,𝑞+𝑐𝐮^𝑗_ℎ,𝑞=0,∀𝑞∈𝐻_ℎ,,𝐮𝑗=1,2,…,𝑁_0ℎ=0,𝑝⁰_ℎ=0 (36) of (24) exists unique zero solution since the number of unknowns is the same as the number of line equations.
By letting 𝐰=𝐮^𝑗_ℎ in the first formula of (36) and 𝑞=𝑝^𝑗_ℎ in the second formula of (36), using Lemma 2, the sum of (36) gives 𝜕_𝑡𝑝^𝑗_ℎ,𝑝^𝑗_ℎ+𝑎𝐮^𝑗_ℎ,𝐮^𝑗_ℎ=0,𝑗=1,2,…,𝑁. (37) Using Lemma 4 and 𝜕_𝑡𝑝^𝑗_ℎ,𝑝^𝑗_ℎ=1𝑝Δ𝑡^𝑗ℎ−𝑝_ℎ^𝑗−1,𝑝^𝑗_ℎ=1𝑝2Δ𝑡^𝑗ℎ,𝑝^𝑗_ℎ−𝑝_ℎ^𝑗−1,𝑝_ℎ^𝑗−1+𝑝^𝑗_ℎ−𝑝_ℎ^𝑗−1,𝑝^𝑗_ℎ−𝑝_ℎ^𝑗−1=1𝑝2Δ𝑡^𝑗ℎ2−𝑝_ℎ^𝑗−12+𝑝^𝑗_ℎ−𝑝_ℎ^𝑗−12≥1𝑝2Δ𝑡^𝑗ℎ2−𝑝_ℎ^𝑗−12, (38) we have, from (37), 𝑝^𝑗_ℎ²−𝑝_ℎ^𝑗−12+2𝐶₀Δ𝑡𝐮^𝑗_ℎ²_div≤0,𝑗=1,2,…,𝑁. (39) Adding the above inequality with 𝑗 from 1 to 𝑖, using 𝑝⁰_ℎ=0, we have 𝑝^𝑖_ℎ²+2𝐶₀Δ𝑡^𝑖_𝑗=1𝐮^𝑗_ℎ²_div≤0,(𝑖=1,2,…,𝑁). (40) Hence, we have ‖𝑝^𝑖_ℎ‖²=0 and |||𝐮^𝑗_ℎ|||²_div=0(𝑖=1,2,…,𝑁); that is, 𝑝^𝑖_ℎ=0 and 𝐮^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 ∀𝑣∈𝐻¹(𝐾), _𝑒𝑖𝜋_𝐾𝑣𝑑𝑠=_𝑒𝑖𝑣𝑑𝑠,(𝑖=1,2,3), (41) where 𝑒_𝑖(𝑖=1,2,3) are the three sides of the element 𝐾∈𝒯_ℎ. It was proved in [5] that 𝜋_𝐾𝑣−𝑣_𝑠,𝐾≤ℎ^2−𝑠|𝑣|_2,𝐾,∀𝑣∈𝐻²(𝐾),(𝑠=0,1,2). (42)

For any 𝐯∈𝐻¹₀(Ω)², define Π₁𝐯∈𝑉_ℎ by Π₁𝐯_𝑖𝐾=Π_𝐾𝑣_𝑖,∀𝐾∈𝒯_ℎ,(𝑖=1,2). (43) Using the definition of Π₁ and integration by parts, we can show that 𝑐𝐯−Π₁𝐯,𝑞=0,∀𝑞∈𝐻_ℎ. (44) It was proved in [6] that 𝐮−Π₁𝐮²_div≤𝑐ℎ²𝐮²₂. (45) 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 𝐮_ℎ,𝑝_𝑛∶(0,𝑡)→𝑉_ℎ×𝐻_ℎ, such that 𝑎𝐮−𝐮_ℎ,𝐰+𝑏𝐰,𝑝−𝑝_ℎ=0,∀𝐰∈𝑉_ℎ,𝑐𝐮−𝐮_ℎ,𝑞=0,∀𝑞∈𝐻_ℎ. (46) It was proved in [15] that (46) has a unique solution and the error estimates in Theorem 8.

Theorem 8. Let (𝐮_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝐻_ℎ be the solution of (46) and let (𝐮,𝑝)∈𝐻²(Ω)²×𝐻¹(Ω) be the solution of (21). Then there exists a positive constant 𝐶 independent of ℎ such that 𝐮−𝐮_ℎdiv+𝑝−𝑝_ℎ≤𝐶ℎ‖𝐮‖₂+𝑝₁. (47)

Theorem 9. Let (𝐮_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝐻_ℎ be the solution of (46) and let (𝐮,𝑝)∈𝐻²(Ω)²×𝐻¹(Ω) be the solution of (21). Then there exists a positive constant 𝐶 independent of ℎ such that 𝐮−𝐮_ℎ𝑡div+𝑝−𝑝_ℎ𝑡≤𝐶ℎ𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁. (48)

Differentiating each equation of (46) on 𝑡 and using (44), (45), 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 10. Let (𝐮_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝐻_ℎ be the solution of (22) and let 𝑝_ℎ(0)=𝑝_ℎ(0),  (𝐮,𝑝)∈𝐻²(Ω)²×𝐻¹(Ω) be the solution of (3). Then there exists a positive constant 𝐶 independent of ℎ such that 𝐮−𝐮_ℎdiv+𝑝−𝑝_ℎ≤𝐶ℎ^𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏+‖𝐮‖₂+𝑝₁. (49)

Proof. Let 𝜉=𝑝_ℎ−𝑝_ℎ, 𝜂=𝐮_ℎ−𝐮_ℎ. Subtracting the two equations of (22) from those of (21), respectively, we have 𝑎𝐮−𝐮_ℎ,𝐰+𝑏𝐰,𝑝−𝑝_ℎ=0,∀𝐰∈𝑉_ℎ,𝑝_𝑡−𝑝_ℎ𝑡,𝑞+𝑐𝐮−𝐮_ℎ,𝑞=0,∀𝑞∈𝐻_ℎ. (50) Using (46), we have 𝑎𝜂,𝐰+𝑏(𝐰,𝜉)=0,∀𝐰∈𝑉_ℎ,𝑝_𝑡−𝑝_ℎ𝑡,𝑞+𝑐𝜂,𝑞=0,∀𝑞∈𝐻_ℎ. (51) Differentiating the first equation of (51) on 𝑡, we have 𝑎𝜂_𝑡,𝐰+𝑏𝐰,𝜉_𝑡=0,∀𝐰∈𝑉_ℎ. (52)
By letting 𝑞=𝜉_𝑡 in the second formula of (51) and letting 𝐰=𝜂 in (52), using Lemma 2, the sum of them gives 𝑎𝜂_𝑡,𝜂+𝜉_𝑡,𝜉_𝑡=𝑝_ℎ−𝑝_𝑡,𝜉_𝑡. (53) Using 𝑎𝜂_𝑡=1,𝜂2𝑑𝑎−1𝑑𝑡𝜂,𝜂2𝑀_𝑡𝜂,𝜂, (54) we have 12𝑑𝑎+𝜉𝑑𝑡𝜂,𝜂_𝑡²≤𝐶𝜉_𝑡⋅𝑝_ℎ−𝑝_𝑡+𝐶𝜂²_div. (55) Multiplying the equation above with 2, integrating them from 0 to 𝑡, and using 𝜖-inequality, Lemma 4, and (48), we can get 𝜂²_div+2𝐶₀^𝑡₀𝜉_𝑡𝑑𝜏²≤𝐶₁ℎ^𝑡₀𝜉_𝑡⋅𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏+𝐶^𝑡₀𝜂²_div≤1𝑑𝑡𝐶₀^𝑡₀𝜉_𝑡𝑑𝜏²+𝐶ℎ^2𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏²+𝐶^𝑡₀𝜂²_div𝑑𝑡; (56) then 𝜂²_div+1𝐶₀^𝑡₀𝜉_𝑡𝑑𝜏²≤𝐶ℎ^2𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏², (57) so 𝜂_div≤𝐶ℎ^𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏,_𝑡0𝜉_𝑡𝑑𝜏≤𝐶ℎ^𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏; (58)(59) hence 𝜉≤^𝑡₀𝜉_𝑡𝑑𝜏≤𝐶ℎ^𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏. (60)
Now, using the triangle inequalities (47), (58), and (60), we get 𝐮−𝐮_ℎdiv+𝑝−𝑝_ℎ≤𝐶ℎ^𝑡₀𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝑑𝜏+‖𝐮‖₂+𝑝₁. (61) The proof is complete.

6. Error Estimates for Fully Discrete Method

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

Theorem 11. Let (𝐮,𝑝)∈𝐻²(Ω)²×𝐻¹(Ω) be the solution of (3), and let (𝐮^𝑗_ℎ,𝑝^𝑗_ℎ)∈𝑉_ℎ×𝐻_ℎ(𝑗=1,2,…,𝑁) be the solution of (24) with 𝑡=𝑡^𝑗(𝑗=1,2,…,𝑁), respectively. If 𝑝⁰_ℎ=𝑝_ℎ(0)=𝑝₀, 𝑢⁰_ℎ=𝑢_ℎ(0)=𝑢₀, then there exists a positive constant C independent of ℎ and Δ𝑡 such that max_{0≤𝑖≤𝑁}𝑝^𝑖−𝑝^𝑖_ℎ≤𝐶Δ𝑡𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}. (62)

Proof. Subtracting the two equations of (24) from (21), respectively, with 𝑡=𝑡^𝑗(𝑗=0,1,…,𝑁), we can get the error equation: 𝑎𝐮^𝑗−𝐮^𝑗_ℎ,𝐰+𝑏𝐰,𝑝^𝑗−𝑝^𝑗_ℎ=0,∀𝐰∈𝑉_ℎ,𝑝^𝑗_𝑡−𝜕_𝑡𝑝^𝑗_ℎ,𝑞+𝑐𝐮^𝑗−𝐮^𝑗_ℎ,𝑞=0,∀𝑞∈𝐻_ℎ,𝑗=1,2,…,𝑁. (63)
Choosing 𝐰=𝜂^𝑗(𝑗=1,2,…,𝑁) and 𝑞=𝜉^𝑗(𝑗=1,2,…,𝑁) in the two equations of (63), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection, we have 𝑎𝜂^𝑗,𝜂^𝑗+𝜕_𝑡𝜉^𝑗,𝜉^𝑗=𝜕_𝑡𝑝^𝑗−𝑝^𝑗_𝑡,𝜉^𝑗+𝜕_𝑡𝜁^𝑗,𝜉^𝑗. (64)
First, we estimate the left item of (64). 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. (65)
Then, we estimate the right item of (64). From 𝜕_𝑡𝑝^𝑗−𝑝^𝑗_𝑡²=1Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝑡^𝑗−1−𝑡𝑝_𝑡𝑡𝑑𝑡²≤_Ω1Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝑡^𝑗−1−𝑡𝑝_𝑡𝑡𝑑𝑡²≤1𝑑𝜒(Δ𝑡)^{2Ω𝑡𝑗𝑡𝑗−1}𝑡^𝑗−1−𝑡²𝑑𝑡^𝑡𝑗_𝑡𝑗−1𝑝²_𝑡𝑡()𝑑𝑡𝑑𝜒≤𝐶Δ𝑡²𝑝_𝑡𝑡²_{𝐿∞(𝐿2)}, (66) we have 𝜕_𝑡𝑝^𝑗−𝑝^𝑗_𝑡,𝜉^𝑗≤𝐶(Δ𝑡)²𝑝_𝑡𝑡²_𝐿∞𝐿2+𝐶𝜉^𝑗2,𝜕_𝑡𝜁^𝑗=𝜁^𝑗−𝜁^𝑗−1=1Δ𝑡Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝜁_𝑡≤1𝑑𝑡Δ𝑡^𝑡𝑗_𝑡^𝑗−1𝜁_𝑡≤𝑑𝑡𝐶ℎΔ𝑡^𝑡𝑗_𝑡^𝑗−1𝐮_𝑡2+𝑝_𝑡1+‖𝐮‖₂+𝑝₁𝐮𝑑𝑡≤𝐶ℎ_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}; (67) therefore 𝜕_𝑡𝜁^𝑗,𝜉^𝑗≤𝐶ℎ²𝐮_𝑡²_{𝐿∞(𝐻2)}+𝑝_𝑡²_{𝐿∞(𝐻1)}+‖𝐮‖²_{𝐿∞(𝐻2)}+𝑝²_{𝐿∞(𝐻1)}²+𝐶𝜉^𝑗2. (68) Substituting the estimations above into (64), using 𝜉⁰=0, we have 𝜉^𝑖2+2𝐶₀Δ𝑡^𝑖_𝑗=1𝜂^𝑗2_div≤𝐶(Δ𝑡)²𝑝_𝑡𝑡²_{𝐿∞(𝐿2)}+𝐶Δ𝑡^𝑖_𝑗=1𝜉^𝑗2+𝐶ℎ²𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}². (69) By 𝜖-inequality and the discrete Gronwall inequality, we have max_{0≤𝑖≤𝑁}𝜉^𝑖≤𝐶Δ𝑡𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}. (70) From the above formula and (47) and using the triangle inequality, we have max_{0≤𝑖≤𝑁}𝑝^𝑖−𝑝^𝑖_ℎ≤𝐶Δ𝑡𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}. (71) This completes the proof.

Theorem 12. Let (𝐮,𝑝)∈𝐻²(Ω)²×𝐻¹(Ω) be the solution of (3), and let (𝐮^𝑗_ℎ,𝑝^𝑗_ℎ)∈𝑉_ℎ×𝐻_ℎ(𝑗=1,2,…,𝑁) be the solution of (24) with 𝑡=𝑡^𝑗(𝑗=1,2,…,𝑁), respectively. If 𝑝⁰_ℎ=𝑝_ℎ(0)=𝑝₀, 𝑢⁰_ℎ=𝑢_ℎ(0)=𝑢₀, then there exists a positive constant C independent of ℎ and Δ𝑡 such that max_{0≤𝑖≤𝑁}𝐮^𝑖−𝐮^𝑖_ℎdiv≤𝐶Δ𝑡𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}. (72)

Proof. Choosing 𝐰=𝜂^𝑗(𝑗=1,2,…,𝑁) and 𝑞=𝜕_𝑡𝜉^𝑗(𝑗=1,2,…,𝑁) in the two equations of (63), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection, we have 𝑎𝜕_𝑡𝜂^𝑗,𝜂^𝑗+𝜕_𝑡𝜉^𝑗,𝜕_𝑡𝜉^𝑗=𝜕_𝑡𝑝^𝑗−𝑝^𝑗_𝑡,𝜕_𝑡𝜉^𝑗+𝜕_𝑡𝜁^𝑗,𝜕_𝑡𝜉^𝑗. (73) From 𝑎𝜕_𝑡𝜂^𝑗,𝜂^𝑗=12𝜕_𝑡𝑎𝜂^𝑗,𝜂^𝑗+Δ𝑡2𝑎𝜕_𝑡𝜂^𝑗,𝜕_𝑡𝜂^𝑗≥𝐶₀𝜂^𝑗2_div (74) and (66) and (67), we have 𝜂^𝑗2_div≤𝐶(Δ𝑡)²𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ²𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}²; (75) therefore 𝜂^𝑗_div≤𝐶Δ𝑡𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}². (76) From the above formula and (47) and using the triangle inequality, we have max_{0≤𝑖≤𝑁}𝐮^𝑖−𝐮^𝑖_ℎdiv≤𝐶Δ𝑡𝑝_{𝑡𝑡𝐿∞(𝐿2)}+𝐶ℎ𝐮_{𝑡𝐿∞(𝐻2)}+𝑝_{𝑡𝐿∞(𝐻1)}+‖𝐮‖_𝐿^∞_(𝐻²₎+𝑝_{𝐿∞(𝐻1)}. (77) The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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).