A Defect-Correction Mixed Finite Element Method for Stationary Conduction-Convection Problems

Abstract

A defect-correction mixed finite element method (MFEM) for solving the stationary conduction-convection problems in two-dimension is given. In this method, we solve the nonlinear equations with an added artificial viscosity term on a grid and correct this solution on the same grid using a linearized defect-correction technique. The stability is given and the error analysis in 𝐿² and 𝐻¹-norm of 𝑢, 𝑇 and the 𝐿²-norm of 𝑝 are derived. The theory analysis shows that our method is stable and has a good precision. Some numerical results are also given, which show that the defect-correction MFEM is highly efficient for the stationary conduction-convection problems.

1. Introduction

In this paper, we consider the stationary conduction-convection problems in two dimension whose coupled equations governing viscous incompressible flow and heat transfer for the incompressible fluid are Boussinesq approximations to the stationary Navier-Stokes equations.

(𝒫) Find (𝑢,𝑝,𝑇)∈𝑋×𝑀×𝑊 such that−𝜈Δ𝑢−𝑢+Δ=𝑇(𝑢d+0⋅i,v∇𝜆𝑢𝑢)𝑇=𝑢⋅∇=+0𝑇,𝑇∇₀=𝑝𝑥0=∈,𝜆Ω𝑗𝑥𝑇,,∈Ω𝑥,∈Ω,,𝑥∈𝜕Ω,(1.1) where Ω is a bounded domain in ℝ² assumed to have a Lipschitz continuous boundary 𝜕Ω. 𝑢=(𝑢₁(𝑥),𝑢₂(𝑥))^𝑇 represents the velocity vector, 𝑝(𝑥) the pressure, 𝑇(𝑥) the temperature, 𝜆>0 the Grashoff number, 𝑗=(0,1)^𝑇 the two-dimensional vector, and 𝜈>0 the viscosity.

As we know the conduction-convection problem contains the velocity vector field, the pressure field and the temperature field, so finding the numerical solution of conduction-convection problems is very difficult. The conduction-convection problems is an important system of equations in atmospheric dynamics and dissipative nonlinear system of equations, so lots of works are devoted to this problem [1–6]. There are also some works devoted to the nonstationary conduction-convection problems [7–10]. In [8], Luo et al. gave an optimizing reduced PLSMFE for the nonstationary conduction-convection problems. They combined PLSMEF method with POD to deal with the problems. In [11], an analysis of conduction natural convection conjugate heat transfer in the gap between concentric cylinders under solar irradiation was studied. In [12], a Newton iterative mixed finite element method for the stationary conduction-convection problems was shown by Si et al. In [13], Si and He gave a coupled Newton iterative mixed finite element method for the stationary conduction-convection problems.

The defect-correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Due to its good efficiency, there are many works devoted to this method, for example, [14–28]. In [18], a method making it possible to apply the idea of iterated defect correction to finite element methods was given. A method for solving the time-dependent Navier-Stokes equations, aiming at higher Reynolds' number, was presented in [23]. In [27], an accurate approximations for self-adjoint elliptic eigenvalues was presented. In [28], Stetter exposed the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.

In this paper we present a defect-correction MFEM for the stationary conduction convection problems. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct this solution on the same grid using a linearized defect-correction technique. Actually, the defect-correction MFEM incorporates the artificial viscosity term as a stabilizing factor, making both the nonlinear system easier to resolve and the linearized system easier to precondition. The stability and error analysis of the coupled the defect-correction MFEM show that this method is stable and has a good precision. Some numerical experiments show that our analysis is proper and our method is effective. And it can be used for solving the convection-conduction problems with much small viscosity.

This paper is organized as follows. In Section 2, the functional settings and some assumptions are given. Section 3 is devoted to the defect-correction MFEM. Section 4 gives the stability analysis. Section 5 presents the error analysis. In Section 6, some numerical results and the numerical analysis to validate the effectiveness of the method are laid out.

2. Functional Setting for the Conduction Convection Problems

In this section, we aim to describe some of the notations and results which will be frequently used in this paper. The Sobolev spaces used in this context are standard [29]. For the mathematical setting of the conduction-convection problems and MFEM of conduction-convection problems (1.1), we introduce the Hilbert spaces 𝑀=𝐿𝑋₂₀=𝐻¹⁰(Ω)²,𝑊=𝐻¹(Ω),(Ω)≐𝜑∈𝐿²(Ω);_Ω𝜑𝑑𝑥=0.(2.1)ℑ_ℎ is the uniformly regular family of triangulation of Ω, indexed by a parameter ℎ=max_{𝐾∈ℑℎ}{ℎ_𝐾;ℎ_𝐾=diam(𝐾)}. We introduce the finite element subspace 𝑋_ℎ⊂𝑋, 𝑀_ℎ⊂𝑀, 𝑊_ℎ⊂𝑊 as follows 𝑋_ℎ=𝑣_ℎ∈𝑋∩𝐶⁰Ω²;𝑣_ℎ|_𝐾∈𝑃_ℓ(𝐾)²,∀𝐾∈ℑ_ℎ𝑀_ℎ,𝑊_ℎ=𝑞_ℎ∈𝑀∩𝐶⁰Ω;𝑞_ℎ|_𝐾∈𝑃_𝑘(𝐾),∀𝐾∈ℑ_ℎ,=𝜙_ℎ∈𝑊∩𝐶⁰Ω;𝜙_ℎ|_𝐾∈𝑃_𝑙(𝐾),∀𝐾∈ℑ_ℎ,(2.2) where 𝑃_ℓ(𝐾) is the space of piecewise polynomials of degree ℓ on 𝐾, and ℓ⩾1, 𝑘⩾1, 𝑙⩾1 are three integers. 𝑊_0ℎ=𝑊_ℎ∩𝐻¹⁰(Ω), and (𝑋_ℎ,𝑀_ℎ) satisfies the discrete LBB conditionsup_{𝑣ℎ∈𝑋ℎ}𝑑𝜑_ℎ,𝑣_ℎ‖‖∇𝑣_ℎ‖‖₀⩾𝛽‖‖𝜑_ℎ‖‖₀,∀𝜑_ℎ∈𝑀_ℎ,(2.3) where 𝑑(𝜑,𝑣)=(𝜑,div𝑣).

With the above notations, the Galerkin mixed variation and the mixed FEM problem for the conduction-convection problems (𝒫) are defined, respectively, as follows.

(𝒫^) Find (𝑢,𝑝,𝑇)∈𝑋×𝑀×𝑊 such that𝜈𝑎(𝑢,𝑣)−𝑑(𝑝,𝑣)+𝑑(𝜑,𝑢)+𝑏(𝑢,𝑢,𝑣)=𝜆(𝑗𝑇,𝑣),∀𝑣∈𝑋,𝜑∈𝑀,𝑎(𝑇,𝜓)+𝜆𝑏(𝑢,𝑇,𝜓)=0,∀𝜓∈𝑊₀.(2.4)

(𝒫^) Find (𝑢_ℎ,𝑝_ℎ,𝑇_ℎ)∈𝑋_ℎ×𝑀_ℎ×𝑊_ℎ such that𝜈𝑎𝑢_ℎ,𝑣_ℎ−𝑑𝑝_ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑢_ℎ+𝑏𝑢_ℎ,𝑢_ℎ,𝑣_ℎ=𝜆𝑗𝑇_ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝑇_ℎ,𝜓_ℎ+𝜆𝑏𝑢_ℎ,𝑇_ℎ,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ,(2.5) where 𝑎(𝑢,𝑣)=(∇𝑢,∇𝑣), 𝑑(𝜑,𝑣)=(𝜑,div𝑣), 𝑎(𝑇,𝜓)=(∇𝑇,∇𝜓), and 𝑏(𝑢,𝑣,𝑤)=12_{Ω2𝑖,𝑘=1}𝑢_𝑖𝜕𝑣_𝑘𝜕𝑥_𝑖𝑤_𝑘𝑑𝑥−²_𝑖_,𝑘=1𝑢_𝑖𝜕𝑤_𝑘𝜕𝑥_𝑖𝑣_𝑘𝑑𝑥,∀𝑢,𝑣,𝑤∈𝑋,𝑏(𝑢,𝑇,𝜓)=12_Ω2𝑖=1𝑢_𝑖𝜕𝑇𝜕𝑥_𝑖𝜓𝑑𝑥−²_𝑖𝑢_𝑖𝜕𝜓𝜕𝑥_𝑖𝑇𝑑𝑥,∀𝑢∈𝑋,𝑇,𝜓∈𝑊.(2.6)

The following assumptions and results are recalled (see [7, 29–31]).

(A₁)There exists a constant 𝐶₀ which only depends on Ω, such that(i)‖𝑢‖₀≤𝐶₀‖∇𝑢‖₀, ‖𝑢‖_0,4≤𝐶₀‖∇𝑢‖₀, for all 𝑢∈𝐻¹⁰(Ω)²(or𝐻¹⁰(Ω)),(ii)‖𝑢‖_0,4≤𝐶₀‖𝑢‖₁,for all 𝑢∈𝐻¹(Ω)²,(iii)‖𝑢‖_0,4≤√2‖∇𝑢‖^01/2‖𝑢‖^01/2,for all𝑢∈𝐻¹⁰(Ω)²(or𝐻¹⁰(Ω)).(A₂)Assuming 𝜕Ω∈𝐶^𝑘,𝛼(𝑘⩾0,𝛼>0), then, for 𝑇₀∈𝐶^𝑘,𝛼(𝜕Ω), there exists an extension in 𝐶^0𝑘,𝛼(ℝ²) (denote 𝑇₀ also), such that ‖‖𝑇₀‖‖_𝑘,𝑞≤𝜀,𝑘⩾0,1≤𝑞≤∞,(2.7) where 𝜀 is an arbitrary positive constant.(A₃)𝑏(⋅,⋅,⋅) and 𝑏(⋅,⋅,⋅) have the following properties.(i)For all 𝑢∈𝑋, 𝑣,𝑤∈𝑋(or𝑇,𝜓∈𝐻¹⁰(Ω)), there holds that 𝑏(𝑢,𝑣,𝑣)=0,𝑏(𝑢,𝑣,𝑤)=−𝑏(𝑢,𝑤,𝑣),(2.8)𝑏(𝑢,𝑇,𝑇)=0,𝑏(𝑢,𝑇,𝜓)=−𝑏(𝑢,𝜓,𝑇).(2.9)(ii)For all𝑢∈𝑋, 𝑣∈𝐻¹(Ω)²(or𝑇∈𝐻¹(Ω)), for all𝑤∈𝑋(or𝜓∈𝐻¹⁰(Ω)), there holds that |||||𝑏(𝑢,𝑣,𝑤)||≤𝑁‖∇𝑢‖₀‖∇𝑣‖₀‖∇𝑤‖₀,(2.10)𝑏(𝑢,𝑇,𝜓)|||≤𝑁‖∇𝑢‖₀‖∇𝑇‖₀‖∇𝜓‖₀,(2.11) where 𝑁=sup^{𝑢,𝑣,𝑤}||𝑏(𝑢,𝑣,𝑤)||‖∇𝑢‖₀‖∇𝑣‖₀‖∇𝑤‖₀,𝑁=sup^{𝑢,𝑇,𝜑}|||𝑏(𝑢,𝑇,𝜑)|||‖∇𝑢‖₀‖∇𝑇‖₀‖∇𝜑‖₀.(2.12)

We recall the following existence, uniqueness and regularity result of (𝒫^) (see [7, Chapter 4]).

Theorem 2.1 (see [7]). Under the assumption of (A₁)~(A₃), letting 𝐴≡2𝜈⁻¹𝜆(3𝐶₀+1)‖𝑇₀‖₁, 𝐵≡2‖∇𝑇₀‖₀+2(𝐶²⁰𝜆)⁻¹𝐴, there exist 0<𝛿₁, 𝛿₂≤1 such that 𝜈⁻¹𝑁𝐴≤1−𝛿₁,𝛿¹⁻¹𝜈⁻¹𝐶²⁰𝜆²𝐵𝑁≤1−𝛿₂.(2.13) Then, there exists a unique solution (𝑢,𝑝,𝑇)∈𝑋×𝑀×𝑊 for (𝒫^), and ‖∇𝑢‖₀≤𝐴,‖∇𝑇‖₀≤𝐵.(2.14)

Some estimates of the trilinear form 𝑏 are given in the following lemma and the proof can be found in [30, 32–34].

Lemma 2.2. The trilinear form 𝑏 satisfies the following estimate: ||𝑏𝑢_ℎ,𝑣_ℎ,𝑤||+||𝑏𝑣_ℎ,𝑢_ℎ,𝑤||+||𝑏𝑤,𝑢_ℎ,𝑣_ℎ||≤𝐶₀||logℎ||^1/2‖‖∇𝑣_ℎ‖‖₀‖‖∇𝑢_ℎ‖‖₀‖𝑤‖₀,(2.15) for all 𝑢_ℎ,𝑣_ℎ∈𝑉_ℎ, 𝑤∈𝐿²(Ω)².

Lemma 2.3. Suppose that (A₁)~(A₃) are valid and 𝜀 is a positive constant, such that 32𝐶²⁰𝜆²𝑁𝜀3𝜈<1,‖‖∇𝑇⁰‖‖₀≤𝜀4,‖‖𝑇⁰‖‖₀≤𝐶⁰𝜀4,(2.16) then (𝒫^) has a unique solution (𝑢_ℎ,𝑝_ℎ,𝑇_ℎ)∈𝑋_ℎ×𝑀_ℎ×𝑊_ℎ, such that 𝑇|_𝜕Ω=𝑇₀ and ‖‖∇𝑢_ℎ‖‖₀≤5𝐶²⁰𝜆𝜀3𝜈,‖‖∇𝑇^ℎ‖‖₀≤𝜀.(2.17)

Proof. The proof of the existence and the uniqueness of the solution has been given by Luo [7]. Let 𝑇_ℎ=𝜔_ℎ+𝑇₀, 𝜓_ℎ=𝜔_ℎ in (2.5), we can get 𝑎𝜔_ℎ,𝜔_ℎ=−𝜆𝑏𝑢_ℎ,𝑇₀,𝜔_ℎ−𝑎𝑇₀,𝜔_ℎ.(2.18) Using (2.11) and (2.16), we deduce ‖‖∇𝜔_ℎ‖‖₀≤‖‖∇𝑇₀‖‖₀+𝜆𝑁𝜀‖‖∇𝑢_ℎ‖‖₀.(2.19) Letting 𝑣_ℎ=𝑢_ℎ, 𝜑_ℎ=𝑝_ℎ in the first equation of (2.5), we get 𝜈‖‖∇𝑢_ℎ‖‖²⁰=||𝜆𝑗𝑇_ℎ,𝑢_ℎ||≤𝜆𝐶₀‖‖𝑇_ℎ‖‖₀‖‖∇𝑢_ℎ‖‖₀.(2.20) By (2.16), we can obtian ‖‖∇𝑢_ℎ‖‖₀≤𝜈⁻¹𝜆𝐶₀‖‖𝑇_ℎ‖‖₀≤≤𝜈𝜈⁻¹₋₁𝜆𝐶₀‖‖𝜔_ℎ‖‖₀+‖‖𝑇₀‖‖₀𝜆𝐶²⁰‖‖∇𝜔_ℎ‖‖₀+𝜈⁻¹𝜆𝐶₀‖‖𝑇₀‖‖₀≤𝜈⁻¹𝜆𝐶₀‖‖𝑇₀‖‖₀+𝜈⁻¹𝜆𝐶²⁰‖‖∇𝑇₀‖‖₀+𝜈⁻¹𝜆²𝐶²⁰𝑁𝜀‖‖∇𝑢_ℎ‖‖₀.(2.21) Using (2.16) again, we get ‖‖∇𝑢_ℎ‖‖₀≤5𝐶²⁰𝜆𝜀3𝜈.(2.22) By (2.19), we deduce ‖‖∇𝑇_ℎ‖‖₀≤‖‖∇𝜔_ℎ‖‖₀+‖‖∇𝑇₀‖‖₀≤2‖‖∇𝑇₀‖‖₀+𝜆𝑁𝜀‖‖∇𝑢_ℎ‖‖₀≤2‖‖∇𝑇₀‖‖₀+5𝐶²⁰𝜆²𝑁𝜀²≤𝜀3𝜈2+2𝜀=𝜀.(2.23)

We introduce the Laplace operator𝒜𝑢=−Δ𝑢,∀𝑢∈𝐷(𝒜)=𝐻²(Ω)²∩𝑋.(2.24)

Lemma 2.4 (see [35, 36]). For all 𝑢,𝑤∈𝑋, 𝑣∈𝐷(𝐴) there holds that ||𝑏(𝑢,𝑣,𝑤)||+||𝑏(𝑣,𝑢,𝑤)||+||𝑏(𝑤,𝑢,𝑣)||≤𝐶‖𝒜𝑣‖₀‖𝑤‖₀‖∇𝑢‖₀.(2.25)

3. The Defect-Correction Method

The aim of this section is to give a method for solving the nonlinear system (2.5) on a coarser mesh than one uses when employing the standard FEM; the coarse-mesh solution is corrected using the same grid in our method. The defect-correction method in which we consider incorporates an artificial viscosity parameter 𝜎ℎ as a stabilizing factor in the solution algorithm. For a fixed grid parameter ℎ the method requires the solution of one nonlinear system and a few linear correction steps. It is described in the following paragraphs. We consider the following problems which is identical to (2.5) except for an artificial viscosity term.

(𝒫^∗) Find (𝑢^0ℎ,𝑝^0ℎ,𝑇^0ℎ)∈𝑋_ℎ×𝑀_ℎ×𝑊_ℎ such that(𝜈+𝜎ℎ)𝑎𝑢^0ℎ,𝑣_ℎ−𝑑𝑝^0ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑢^0ℎ∀𝑣+_ℎ𝑏𝑢^0ℎ,𝑢^0ℎ,𝑣_ℎ=𝜆𝑗𝑇^0ℎ,𝑣_ℎ,(1+𝜎ℎ)∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝑇^0ℎ,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ.(3.1) We define the residual or named defect 𝑅(𝑢^0ℎ,𝑝^0ℎ,𝑇^0ℎ), 𝑄(𝑢^0ℎ,𝑝^0ℎ,𝑇^0ℎ) for the momentum systems as follows:𝑅𝑢^0ℎ,𝑝^0ℎ,𝑇^0ℎ,𝑣_ℎ=−𝜆𝑑𝑗𝑇𝜑^0ℎ_ℎ,𝑣_ℎ−𝜈𝑎𝑢^0ℎ,𝑣_ℎ+𝑑𝑝^0ℎ,𝑣_ℎ𝑄𝑢_0ℎ,𝑢^0ℎ−𝑏𝑢^0ℎ,𝑢^0ℎ,𝑣_ℎ,,𝑝^0ℎ,𝑇^0ℎ,𝜓_ℎ=−𝑎𝑇^0ℎ,𝜓_ℎ−𝜆𝑏𝑢^0ℎ,𝑇^0ℎ,𝜓_ℎ.(3.2) Define the correction (𝜀^0ℎ,𝜚^0ℎ,𝜏^0ℎ) satisfying the following linear problem:(𝜈+=𝜎ℎ)𝑅𝑎𝑢𝜀^0ℎ_0ℎ,𝑣_ℎ−𝑑𝜚^0ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝜀^0ℎ+𝑏𝜀^0ℎ,𝑢^0ℎ,𝑣_ℎ+𝑏𝑢^0ℎ,𝜀^0ℎ,𝑣_ℎ(1+𝜎ℎ),𝑝^0ℎ,𝑇^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,=𝑄𝑎𝑢𝜏^0ℎ_0ℎ,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝜏^0ℎ,𝜓_ℎ+𝜆𝑏𝜀^0ℎ,𝑇^0ℎ,𝜓_ℎ,𝑝^0ℎ,𝑇^0ℎ,𝜓_ℎ,∀𝜓_ℎ∈𝑊_0ℎ.(3.3) Define 𝑢^1ℎ=𝑢^0ℎ+𝜀⁰, 𝑝^1ℎ=𝑝^0ℎ+𝜚^0ℎ, 𝑇^1ℎ=𝑇^0ℎ+𝜏^0ℎ, which are hoped to be better solutions of the problems. In order to obtain the equations for (𝑢^1ℎ,𝑝^1ℎ,𝑇^1ℎ), we use the residual equation (3.2) to rewrite the linear problems (3.3); we obtain𝒫^†⎧⎪⎪⎨⎪⎪⎩(𝜈+𝜎ℎ)𝑎𝑢^1ℎ=𝜆𝑗𝑇_1ℎ,𝑣_ℎ−𝑑𝑝^1ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑢^1ℎ+𝑏𝑢^0ℎ,𝑢^1ℎ,𝑣_ℎ+𝑏𝑢^1ℎ,𝑢^0ℎ,𝑣_ℎ(1+𝜎ℎ),𝑣_ℎ+𝜎ℎ𝑎𝑢^0ℎ,𝑣_ℎ+𝑏𝑢^0ℎ,𝑢^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,=𝜎ℎ𝑎𝑇^1ℎ,𝜓_ℎ+𝜆𝑏𝑢^1ℎ,𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝑇^1ℎ,𝜓_ℎ𝑎𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝑇^0ℎ,𝜓_ℎ,∀𝜓_ℎ∈𝑊_0ℎ.(3.4) In general, this method can be described as follows.

Step 1. Solve the nonlinear systems (3.1) for (𝑢^0ℎ,𝑝^0ℎ,𝑇^0ℎ).

Step 2. For 𝑖=1,2,…,𝑚, solve the linear equations 𝒫^‡⎧⎪⎪⎨⎪⎪⎩(𝜈+𝜎ℎ)𝑎𝑢^𝑖ℎ=𝑇_𝑖ℎ,𝑣_ℎ−𝑑𝑝^𝑖ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑢^𝑖ℎ+𝑏𝑢^ℎ𝑖−1,𝑢^𝑖ℎ,𝑣_ℎ+𝑏𝑢^𝑖ℎ,𝑢^ℎ𝑖−1,𝑣_ℎ(1+𝜎ℎ),𝑣_ℎ+𝜎ℎ𝑎𝑢^ℎ𝑖−1,𝑣_ℎ+𝑏𝑢^ℎ𝑖−1,𝑢^ℎ𝑖−1,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,=𝜎ℎ𝑎𝑇^𝑖ℎ,𝜓_ℎ+𝜆𝑏𝑢^𝑖ℎ,𝑇^ℎ𝑖−1,𝜓_ℎ+𝜆𝑏𝑢^ℎ𝑖−1,𝑇^𝑖ℎ,𝜓_ℎ𝑎𝑇^ℎ𝑖−1,𝜓_ℎ+𝜆𝑏𝑢^ℎ𝑖−1,𝑇^ℎ𝑖−1,𝜓_ℎ,∀𝜓_ℎ∈𝑊_0ℎ.(3.5) For each 𝑖 the residual is given by 𝑅𝑢^𝑖ℎ,𝑝^𝑖ℎ,𝑇^𝑖ℎ,𝑣_ℎ=−𝜆𝑑𝑗𝑇𝜑^𝑖ℎ,_ℎ𝑣_ℎ−𝜈𝑎𝑢^𝑖ℎ,𝑣_ℎ+𝑑𝑝^𝑖ℎ,𝑣_ℎ𝑄𝑢_𝑖ℎ,𝑢^𝑖ℎ−𝑏𝑢^𝑖ℎ,𝑢^𝑖ℎ,𝑣_ℎ,,𝑝^𝑖ℎ,𝑇^𝑖ℎ,𝜓_ℎ=−𝑎𝑇^𝑖ℎ,𝜓_ℎ−𝜆𝑏𝑢^𝑖ℎ,𝑇^𝑖ℎ,𝜓_ℎ.(3.6) The correction (𝜀^𝑖ℎ,𝜚^𝑖ℎ,𝜏^𝑖ℎ) is given by (𝜈+=𝜎ℎ)𝑅𝑎𝑢𝜀^𝑖ℎ_𝑖ℎ,𝑣_ℎ−𝑑𝜚^𝑖ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝜀^𝑖ℎ+𝑏𝜀^𝑖ℎ,𝑢^𝑖ℎ,𝑣_ℎ+𝑏𝑢^𝑖ℎ,𝜀^𝑖ℎ,𝑣_ℎ(1+𝜎ℎ),𝑝^𝑖ℎ,𝑇^𝑖ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,=𝑄𝑎𝑢𝜏^𝑖ℎ_𝑖ℎ,𝜓_ℎ+𝜆𝑏𝑢^𝑖ℎ,𝜏^𝑖,𝜓_ℎ+𝜆𝑏𝜀^𝑖ℎ,𝑇^𝑖ℎ,𝜓_ℎ,𝑝^𝑖ℎ,𝑇^𝑖ℎ,𝜓_ℎ,∀𝜓_ℎ∈𝑊_0ℎ.(3.7)

Remark 3.1. From the numerical experiments, we see that one or two correction steps is adequate. And this is as same as [24].

4. Stability Analysis

In this section, we give the stability analysis. It is given by the following theorems.

Theorem 4.1. Under the assumptions of Lemma 2.3, then (𝑢^0ℎ,𝑇^0ℎ) defined by (𝒫^∗) satisfies ‖‖∇𝑢^0ℎ‖‖₀≤5𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ),‖‖∇𝑇^0ℎ‖‖₀≤𝜀.(4.1) Moreover, if 25𝐶²⁰𝑁𝜆𝜀3(𝜈+𝜎ℎ)²<1,(4.2)(𝒫^∗) admits a unique solution.

Proof. We define the set ℬ_𝑀=𝑣_ℎ̃∈𝑋_ℎ;‖‖∇𝑣_ℎ̃‖‖₀≤5𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ).(4.3)
Let 𝑢_ℎ̃ be in ℬ_𝑀. Then(1+𝜎ℎ)𝑎𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢_ℎ̃,𝑇^0ℎ,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ(4.4) has a unique solution 𝑇^0ℎ∈𝑊_ℎ such that 𝑇_ℎ|_𝜕Ω=𝑇₀. For a given 𝑇^0ℎ, we consider the following problem: (𝜈+𝜎ℎ)𝑎𝑢^ℎ0∗,𝑣_ℎ−𝑑𝑝^ℎ0∗,𝑣_ℎ+𝑑𝜑_ℎ,𝑢^ℎ0∗∀𝑣+_ℎ𝑏𝑢^ℎ0∗,𝑢^ℎ0∗,𝑣_ℎ=𝜆𝑗𝑇^0ℎ,𝑣_ℎ,∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ.(4.5) By the theory of the Navier-Stokes equations, we get (4.5) has a unique solution (𝑢^ℎ0∗,𝑝^ℎ0∗)∈𝑋_ℎ×𝑀_ℎ (see [31]). It means that (4.4) and (4.5) give a unique 𝑢^ℎ0∗∈𝑋_ℎ for a given 𝑢_ℎ̃∈𝑋_ℎ, we denote 𝑢^ℎ0∗=ℓ_ℎ𝑢_ℎ̃.(4.6)
Setting 𝑇^0ℎ=𝜔^0ℎ+𝑇₀, 𝜓_ℎ=𝜔^0ℎ in (4.4) and using (2.9), we can obtain(1+𝜎ℎ)𝑎𝜔^0ℎ,𝜔^0ℎ=−𝜆𝑏𝑢_ℎ̃,𝑇₀,𝜔^0ℎ−(1+𝜎ℎ)𝑎𝑇₀,𝜔^0ℎ.(4.7) Using (2.7), (2.11), and (2.16), we can get (1+𝜎ℎ)‖‖∇‖‖𝜔∇^0ℎ‖‖𝜔_00ℎ≤𝜆𝑁‖‖∇𝑢_ℎ̃‖‖₀‖‖∇𝑇₀‖‖₀+(1+𝜎ℎ)‖‖∇𝑇₀‖‖₀,‖‖₀≤𝜆𝑁𝜀4‖‖∇𝑢^ℎ̃‖‖₀+‖‖∇𝑇₀‖‖₀.(4.8) Using the triangle inequality, we have ‖‖∇𝑇^0ℎ‖‖₀≤‖‖∇𝑇₀‖‖₀+‖‖∇𝜔^0ℎ‖‖₀≤𝜆𝑁𝜀4‖‖∇𝑢^ℎ̃‖‖₀+2‖‖∇𝑇₀‖‖₀≤5𝐶²⁰𝑁𝜆𝜀²12(𝜈+𝜎ℎ)+2𝜀≤𝜀.(4.9)
Letting 𝑣_ℎ=𝑢^ℎ0∗, 𝜑_ℎ=𝑝^0ℎ in (4.5) and using (2.8), we get(𝜈+𝜎ℎ)𝑎𝑢^ℎ0∗,𝑢^ℎ0∗=𝜆𝑗𝑇^0ℎ,𝑢^ℎ0∗.(4.10) Letting 𝑇^0ℎ=𝜔^0ℎ+𝑇₀ and using (2.9), we have (𝜈+𝜎ℎ)‖‖∇𝑢^ℎ0∗‖‖₀≤𝐶²⁰𝜆‖‖∇𝜔^0ℎ‖‖₀+𝐶₀𝜆‖‖𝑇₀‖‖₀≤𝐶²⁰𝜆²𝑁‖‖∇𝑢_ℎ̃‖‖₀‖‖∇𝑇₀‖‖₀+𝐶₀𝜆1+𝐶₀‖‖∇𝑇₀‖‖₀≤𝐶²⁰𝜆𝜀.(4.11) Namely, ‖‖∇𝑢^ℎ0∗‖‖₀≤5𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ).(4.12) Hence, we proved that ℓ_ℎ maps ℬ_𝑀 to ℬ_𝑀. It follows from Brouwer's fixed-point theorem that there exits a solution to system (𝒫^∗).
To prove the uniqueness, assume that (𝑢^ℎ01,𝑝^ℎ01,𝑇^ℎ01),(𝑢^ℎ02,𝑝^ℎ02,𝑇^ℎ02)∈𝑋_ℎ×𝑀_ℎ×𝑊_ℎ, and 𝑇^ℎ01|_𝜕Ω=𝑇^ℎ02|_𝜕Ω=𝑇₀ are two solutions of (𝒫^∗). Then, we obtain that(𝜈+𝜎+ℎ)𝑏𝑎𝑢𝑢^ℎ01_ℎ02−𝑢^ℎ02,𝑣_ℎ−𝑑𝑝^ℎ01−𝑝^ℎ02,𝑣_ℎ+𝑑𝜑_ℎ,𝑢^ℎ01−𝑢^ℎ02+𝑏𝑢^ℎ01−𝑢^ℎ02,𝑢^ℎ01,𝑣_ℎ(1+𝜎ℎ),𝑢^ℎ01−𝑢^ℎ02,𝑣_ℎ=𝜆𝑗𝑇^ℎ01−𝑇^ℎ02,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝑇^ℎ01−𝑇^ℎ02,𝜓_ℎ+𝜆𝑏𝑢^ℎ02,𝑇^ℎ01−𝑇^ℎ02,𝜓_ℎ+𝜆𝑏𝑢^ℎ01−𝑢^ℎ02,𝑇^ℎ01,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ.(4.13) Let 𝑣_ℎ=𝑢^ℎ01−𝑢^ℎ02, 𝜑_ℎ=𝑝^ℎ01−𝑝^ℎ02 in the first equation of (4.13), we can get (𝜈+𝜎ℎ)‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀≤𝑁‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀‖‖∇𝑢^ℎ01‖‖₀+𝐶²⁰𝜆‖‖∇𝑇^ℎ01−𝑇^ℎ02‖‖₀.(4.14) Setting 𝜓_ℎ=𝑇^ℎ01−𝑇^ℎ02 in the second equation of (4.13), we obtain (1+𝜎ℎ)‖‖∇𝑇^ℎ01−𝑇^ℎ02‖‖₀≤𝜆𝑁‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀‖‖∇𝑇^ℎ01‖‖₀.(4.15) By (4.14) and (4.15), we deduce (𝜈+𝜎ℎ)‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀≤𝑁‖‖∇𝑢^ℎ01‖‖₀‖‖∇(𝑢^ℎ01−𝑢^ℎ02)‖‖₀+𝐶²⁰𝜆²𝑁‖‖∇𝑇^ℎ01‖‖₀‖‖∇(𝑢^ℎ01−𝑢^ℎ02)‖‖₀≤5𝐶²⁰𝑁𝜆𝜀3(𝜈+𝜎ℎ)‖‖∇(𝑢^ℎ01−𝑢^ℎ02)‖‖₀+𝐶²⁰𝜆𝑁𝜀4𝜈‖‖∇(𝑢^ℎ01−𝑢^ℎ02)‖‖₀.(4.16) Using (4.2), we obtain ‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀≤25‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀.(4.17) Namely, ‖‖∇𝑢^ℎ01−𝑢^ℎ02‖‖₀=0.(4.18) By (4.15), we see that ‖∇(𝑇^ℎ01−𝑇^ℎ02)‖₀=0. Therefore, it follows that (𝒫^∗) has a unique solution.
Then, we give the prove of (4.1) without using (4.2). Letting 𝑣_ℎ=𝑢^0ℎ, 𝜑_ℎ=𝑝^0ℎ in the first equation of (3.1) and using (2.8), we get(𝜈+𝜎ℎ)𝑎𝑢^0ℎ,𝑢^0ℎ=𝜆𝑗𝑇^0ℎ,𝑢^0ℎ.(4.19) Letting 𝑇^0ℎ=𝜔^0ℎ+𝑇₀, we have (𝜈+𝜎ℎ)‖‖∇𝑢^0ℎ‖‖₀≤𝐶²⁰𝜆‖‖∇𝜔^0ℎ‖‖₀+𝐶₀𝜆‖‖𝑇₀‖‖₀.(4.20) Letting 𝑇^0ℎ=𝜔^0ℎ+𝑇₀, 𝜓_ℎ=𝜔^0ℎ in the second equation of (3.1) and using (2.9), we can obtain (1+𝜎ℎ)𝑎𝜔^0ℎ,𝜔^0ℎ=−𝜆𝑏𝑢^0ℎ,𝑇₀,𝜔^0ℎ−(1+𝜎ℎ)𝑎𝑇₀,𝜔^0ℎ.(4.21) Using (2.7), (2.11), and (2.16), we can get (1+𝜎ℎ)‖‖∇‖‖𝜔∇^0ℎ‖‖𝜔_00ℎ≤𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇₀‖‖₀+(1+𝜎ℎ)‖‖∇𝑇₀‖‖₀,‖‖₀≤𝜆𝑁𝜀4‖‖∇𝑢^0ℎ‖‖₀+‖‖∇𝑇₀‖‖₀.(4.22) By (4.20) and (4.22), we can deduce ‖‖∇𝑢^0ℎ‖‖₀≤(𝜈+𝜎ℎ)⁻¹𝜆𝐶²⁰‖‖∇𝜔^0ℎ‖‖₀+(𝜈+𝜎ℎ)⁻¹𝐶₀𝜆‖‖𝑇₀‖‖₀≤(𝜈+𝜎ℎ)⁻¹𝜆𝐶⁰‖‖𝑇₀‖‖₀+𝐶²⁰𝜆‖‖∇𝑇₀‖‖₀+𝐶²⁰𝜆²𝑁𝜀4‖‖∇𝑢^0ℎ‖‖₀.(4.23) Using (2.16), we get ‖‖∇𝑢^0ℎ‖‖₀≤5𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ).(4.24) Using (2.7), (2.11), (2.16), and (4.20), we can get ‖‖∇𝜔^0ℎ‖‖₀≤𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇₀‖‖₀+‖‖∇𝑇₀‖‖₀≤𝜆𝑁𝜀4‖‖∇𝑢^0ℎ‖‖₀+‖‖∇𝑇₀‖‖₀≤3𝜀‖‖∇𝑇_0ℎ4,‖‖₀≤‖‖∇𝜔^0ℎ‖‖₀+‖‖∇𝑇₀‖‖₀≤𝜀.(4.25) Therefore, we finish the proof.

Theorem 4.2. Under the assumptions of Lemma 2.3, and 25𝐶²⁰𝑁𝜆𝜀3(𝜈+𝜎ℎ)²<1,(4.26)(𝑢^1ℎ,𝑇^1ℎ) defined by (3.4) satisfies ‖‖∇𝑢^1ℎ‖‖₀≤𝛿,‖‖∇𝑇^1ℎ‖‖₀≤5𝜀6+𝜆𝑁𝛿𝜀+𝜎ℎ𝜀,(4.27) where 𝛿≐(103𝐶²⁰𝜆𝜀/48+𝜎ℎ(5𝐶²⁰𝜆𝜀/3𝜈))/(7/10)(𝜈+𝜎ℎ).

Proof. Letting 𝑣_ℎ=𝑢^1ℎ, 𝜑_ℎ=𝑝^1ℎ in the first equation of (3.4) and using (2.8), we get (𝜈+𝜎ℎ)𝑎𝑢^1ℎ,𝑢^1ℎ+𝑏𝑢^1ℎ,𝑢^0ℎ,𝑢^1ℎ=𝑏𝑢^0ℎ,𝑢^0ℎ,𝑢^1ℎ+𝜎ℎ𝑎𝑢^0ℎ,𝑢^1ℎ+𝜆𝑗𝑇^0ℎ,𝑢^1ℎ.(4.28) Letting 𝑇^0ℎ=𝜔^0ℎ+𝑇₀ and using (2.10), we have (𝜈+𝜎ℎ)‖‖∇𝑢^1ℎ‖‖₀≤𝑁‖‖∇𝑢^1ℎ‖‖₀‖‖∇𝑢^0ℎ‖‖₀+𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝑁‖‖∇𝑢^0ℎ‖‖²⁰+𝐶²⁰𝜆‖‖∇𝜔^1ℎ‖‖₀+𝐶₀𝜆‖‖𝑇₀‖‖₀.(4.29) Let 𝑇^1ℎ=𝜔^1ℎ+𝑇₀, 𝜓_ℎ=𝜔^1ℎ in the second equation of (3.4), we can obtain (1+𝜎ℎ)𝑎𝜔^1ℎ,𝜔^1ℎ=+−𝜆𝜎𝑏ℎ𝑢^0ℎ,𝑇₀,𝜔^1ℎ−𝜆𝑏𝑢^1ℎ,𝑇^0ℎ,𝜔^1ℎ+𝜆𝑏𝑢^0ℎ,𝑇^0ℎ,𝜔^1ℎ𝑎𝑇^0ℎ,𝜔^1ℎ−(1+𝜎ℎ)𝑎𝑇₀,𝜔^1ℎ.(4.30) Using (2.11) and (2.16), we can get (1+𝜎ℎ)‖‖∇𝜔^1ℎ‖‖₀≤𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇₀‖‖₀+𝜆𝑁‖‖∇𝑢^1ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀‖‖∇𝜔_1ℎ+𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀+𝜎ℎ‖‖∇𝑇^0ℎ‖‖₀+(1+𝜎ℎ)‖‖∇𝑇₀‖‖₀,(4.31)‖‖₀≤𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇₀‖‖₀+𝜆𝑁‖‖∇𝑢^1ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀+𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀+𝜎ℎ‖‖∇𝑇^0ℎ‖‖₀+‖‖∇𝑇₀‖‖₀.(4.32) Using (4.29), we get (𝜈+𝜎ℎ)‖‖∇𝑢^1ℎ‖‖₀≤𝑁‖‖∇𝑢^1ℎ‖‖₀‖‖∇𝑢^0ℎ‖‖₀+𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝑁‖‖∇𝑢^0ℎ‖‖²⁰+𝐶²⁰𝜆𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇₀‖‖₀+𝜆𝑁‖‖∇𝑢^1ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀+𝜆𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀𝜈+𝜎ℎ−𝑁+‖‖‖‖∇∇𝑢𝑇^0ℎ_0ℎ‖‖₀+‖‖∇𝑇₀‖‖₀+𝐶₀𝜆‖‖𝑇₀‖‖₀.‖‖₀−𝐶²⁰𝜆²𝑁‖‖∇𝑇^0ℎ‖‖₀‖‖∇𝑢^1ℎ‖‖₀≤𝑁‖‖∇𝑢^0ℎ‖‖²⁰+𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝐶²⁰𝜆²𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇₀‖‖₀+𝐶²⁰𝜆²𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀+𝐶²⁰𝜆‖‖∇𝑇^0ℎ‖‖₀+𝐶²⁰𝜆‖‖∇𝑇₀‖‖₀+𝐶₀𝜆‖‖𝑇₀‖‖₀.(4.33) Using (2.16), (4.26), and Theorem 4.2, we can obtain 710(𝜈+𝜎ℎ)‖‖∇𝑢^1ℎ‖‖₀≤25𝑁𝐶⁴⁰𝜆²𝜀²9(𝜈+𝜎ℎ)²+𝜎ℎ5𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ)+10𝑁𝐶⁴⁰𝜆³𝜀²3(𝜈+𝜎ℎ)+3𝐶²⁰𝜆²𝜀²≤103𝐶₂₀2𝜆𝜀48+𝜎ℎ5𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ).(4.34) Namely, ‖‖∇𝑢^1ℎ‖‖₀≤103𝐶²⁰𝜆𝜀/48+𝜎ℎ5𝐶²⁰𝜆𝜀/3𝜈(7/10)(𝜈+𝜎ℎ)≐𝛿.(4.35)
Using (2.16), (4.31), and (4.35), we can get‖‖∇𝜔^1ℎ‖‖₀≤10𝑁𝐶²⁰𝜆²𝜀²3(𝜈+𝜎ℎ)+𝜆𝑁𝛿𝜀+𝜎ℎ𝜀+𝜀≤𝜀43+4𝜀+𝜆=7𝜀𝑁𝛿𝜀+𝜎ℎ𝜀12+𝜆𝑁𝛿𝜀+𝜎ℎ𝜀.(4.36) Using the triangle inequality, we can get ‖‖∇𝑇^1ℎ‖‖₀≤‖‖∇𝜔^1ℎ‖‖₀+‖‖∇𝑇₀‖‖₀≤5𝜀6+𝜆𝑁𝛿𝜀+𝜎ℎ𝜀.(4.37) Therefore, we finish the proof.

5. Error Analysis

In this section, we establish the 𝐻¹ and 𝐿²-bounds of the error 𝑢−𝑢^𝑖ℎ, 𝑇−𝑇^𝑖ℎ, 𝑖=0,1 and 𝐿²-bound of the error 𝑝−𝑝^𝑖ℎ, 𝑖=0,1. In order to obtain the error estimates, we define the Galerkin projection (𝑅_ℎ,𝑄_ℎ)=(𝑅_ℎ(𝑢,𝑝),𝑄_ℎ(𝑢,𝑝))∶(𝑋,𝑀)→(𝑋_ℎ,𝑀_ℎ), such that𝑎𝑅_ℎ−𝑢,𝑣_ℎ−𝑑𝑄_ℎ−𝑝,𝑣_ℎ+𝑑𝑞_ℎ,𝑅_ℎ−𝑢=0,∀(𝑢,𝑝)∈(𝑋,𝑀),𝑣_ℎ,𝑞_ℎ∈𝑋_ℎ,𝑀_ℎ.(5.1)

Lemma 5.1 (see [37, 38]). The Galerkin projection (𝑅_ℎ,𝑄_ℎ) satisfies ‖‖𝑅_ℎ−𝑢‖‖₀+ℎ‖‖∇𝑅_ℎ−𝑢‖‖₀+‖‖𝑄_ℎ−𝑝‖‖₀≤𝐶ℎ^𝑟+1𝜈‖𝑢‖_𝑟+1+‖𝑝‖_𝑟,𝑟=1,2.(5.2)

Lemma 5.2 (see [7]). There exits 𝑟_ℎ̃∶𝑊→𝑊_ℎ for all 𝜓∈𝑊 holds that ∇𝜓−𝑟_ℎ̃𝜓,∇𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_ℎ,(5.3)_Ω𝜓−𝑟_ℎ̃𝜓𝑑𝑥=0,‖‖∇𝑟_ℎ̃𝜓‖‖₀≤‖∇𝜓‖₀.(5.4) When 𝜓∈𝑊^𝑘,𝑞(Ω)(1≤𝑞≤∞), there holds ‖‖𝜓−𝑟_ℎ̃𝜓‖‖_−𝑠,𝑞≤𝐶ℎ^𝑘+𝑠||𝜓||_𝑘,𝑞,−1≤𝑠≤𝑚,0≤𝑘≤𝑟+1.(5.5) There exits 𝑟_ℎ∶𝑊₀→𝑊_0ℎ for all 𝜓∈𝑊₀ holds that ∇𝜓−𝑟_ℎ𝜓,∇𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ,‖‖∇𝑟_ℎ𝜓‖‖₀≤‖∇𝜓‖₀.(5.6) When 𝜓∈𝑊^𝑟,𝑞(Ω)(1≤𝑞≤∞), there holds ‖‖𝜓−𝑟_ℎ𝜓‖‖_−𝑠,𝑞≤𝐶ℎ^𝑘+𝑠||𝜓||_𝑟,𝑞,−1≤𝑠≤𝑟,0≤𝑘≤𝑟+1.(5.7)

Lemma 5.3 (see [7]). If (A₁)~(A₃) hold and (𝑢,𝑝,𝑇)∈𝐻^𝑟+1(Ω)×𝐻^𝑟(Ω)×𝐻^𝑟+1(Ω) and (𝑢_ℎ,𝑃_ℎ,𝑇_ℎ) are the solution of problem (𝒫^) and (𝒫^), respectively, then there holds that ‖‖∇(𝑢−𝑢_ℎ)‖‖₀+‖‖𝑝−𝑝_ℎ‖‖₀+‖‖∇(𝑇−𝑇_ℎ)‖‖₀≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1.(5.8)

Lemma 5.4. Under the assumptions of Lemma 2.3, (𝑢_ℎ,𝑝_ℎ,𝑇_ℎ) is the solution of (3.1), (𝑢^0ℎ,𝑝^0ℎ,𝑇^0ℎ) defined by (3.4), then there hold ‖‖∇(𝑢_ℎ−𝑢^0ℎ)‖‖₀≤50𝜎𝐶²⁰𝜆ℎ21𝜈+10𝐶²⁰𝜆𝜎𝜀ℎ‖‖∇(𝑇_ℎ7𝜈,−𝑇^0ℎ)‖‖₀≤2𝜎ℎ𝜀+𝜎ℎ𝜀𝛽‖‖𝑝14𝜈_ℎ,−𝑝^0ℎ‖‖₀≤95𝜎ℎ𝐶²⁰𝜆𝜀21𝜈+19𝜎ℎ𝐶²⁰𝜆𝜀7+𝜎ℎ𝐶²⁰𝜆+𝜎ℎ𝐶²⁰𝜆14𝜈.(5.9)

Proof. Subtracting (3.1) from (2.5) we get the error equations, namely (𝑢_ℎ−𝑢^0ℎ,𝑝_ℎ−𝑝^0ℎ,𝑇_ℎ−𝑇^0ℎ) satisfy 𝜈𝑎𝑢_ℎ+−𝑏𝑢^0ℎ,𝑢𝑣_ℎℎ−𝜎ℎ𝑎𝑢^0ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑢_ℎ−𝑢^0ℎ−𝑑𝑝_ℎ−𝑝^0ℎ,𝑣_ℎ+𝑏𝑢^0ℎ,𝑢_ℎ−𝑢^0ℎ,𝑣_ℎ−𝑢^0ℎ,𝑢_ℎ,𝑣_ℎ=𝜆𝑗𝑇_ℎ−𝑇^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝑇_ℎ−𝑇^0ℎ,𝜓_ℎ−𝜎ℎ𝑎𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢_ℎ−𝑢^0ℎ,𝑇_ℎ,𝜓_ℎ+𝑏𝑢^0ℎ,𝑇_ℎ−𝑇^0ℎ,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ.(5.10) Letting 𝑣_ℎ=𝑢_ℎ−𝑢^0ℎ,𝜑_ℎ=𝑝_ℎ−𝑃^0ℎ in the first equation of (5.10) and using (2.11), (2.8), and (A₁), we can get 𝜈‖‖∇𝑢_ℎ−𝑢^0ℎ‖‖₀≤𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝑁‖‖∇𝑢_ℎ−𝑢^0ℎ‖‖₀‖‖∇𝑢_ℎ‖‖₀+𝐶²⁰𝜆‖‖∇𝑇_ℎ−𝑇^0ℎ‖‖₀.(5.11) Hence, we deduce 𝜈−𝑁‖‖∇𝑢_ℎ‖‖₀‖‖∇𝑢_ℎ−𝑢^0ℎ‖‖₀≤𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝐶²⁰𝜆‖‖∇𝑇_ℎ−𝑇^0ℎ‖‖₀.(5.12) Letting 𝜓_ℎ=𝑇_ℎ−𝑇^0ℎ in the second equation of (5.10) and using (2.9), we obtain 𝑎𝑇_ℎ−𝑇^0ℎ,𝑇_ℎ−𝑇^0ℎ+𝜎ℎ𝑎𝑇^0ℎ,𝑇_ℎ−𝑇^0ℎ+𝜆𝑏𝑢_ℎ−𝑢^0ℎ,𝑇_ℎ,𝑇_ℎ−𝑇^0ℎ=0.(5.13) Using (2.11), we can get ‖‖∇(𝑇_ℎ−𝑇^0ℎ)‖‖₀≤𝜎ℎ‖‖∇𝑇^0ℎ‖‖₀+𝜆𝑁‖‖∇(𝑢_ℎ−𝑢^0ℎ)‖‖₀‖‖∇𝑇_ℎ‖‖₀.(5.14) By (5.12), we deduce 𝜈−𝑁‖‖∇𝑢_ℎ‖‖₀‖‖∇𝑢_ℎ−𝑢^0ℎ‖‖₀≤𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝐶²⁰𝜆𝜎ℎ‖‖∇𝑇^0ℎ‖‖₀+𝐶²⁰𝜆²𝑁‖‖∇(𝑢_ℎ−𝑢^0ℎ)‖‖₀‖‖∇𝑇_ℎ‖‖₀.(5.15) Using (4.1), we can obtain 𝜈−𝑁‖‖∇𝑢^ℎ‖‖₀−33𝜈2‖‖∇𝑢^ℎ−𝑢^0ℎ‖‖₀≤𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝐶²⁰𝜆𝜎ℎ‖‖∇𝑇^0ℎ‖‖₀≤5𝜎ℎ𝐶²⁰𝜆𝜀3(𝜈+𝜎ℎ)+𝐶²⁰𝜆𝜎ℎ𝜀.(5.16) By using (2.16) and (2.17), there holds 𝜈−𝑁‖‖∇𝑢_ℎ‖‖₀−3𝜈32≥71𝜈0.(5.17) Therefore, we can deduce ‖‖∇(𝑢_ℎ−𝑢^0ℎ)‖‖₀≤50𝜎ℎ𝐶²⁰𝜆𝜀21𝜈(𝜈+𝜎ℎ)+10𝐶²⁰𝜆𝜎ℎ𝜀7𝜈.(5.18) By (5.14) and (5.18), we can have ‖‖∇(𝑇_ℎ−𝑇^0ℎ)‖‖₀≤𝜎ℎ𝜀+𝜆𝑁𝜀50𝜎ℎ𝐶²⁰𝜆𝜀21𝜈(𝜈+𝜎ℎ)+10𝐶²⁰𝜆𝜎ℎ𝜀≤2𝜎ℎ𝜀+𝜎ℎ𝜀7𝜈14𝜈.(5.19) Letting 𝜑_ℎ=0, 𝑣_ℎ=𝑢_ℎ−𝑢^0ℎ in the first equation of (5.10) and using (2.3), we have 𝛽‖‖𝑝_ℎ−𝑝^0ℎ‖‖₀≤𝜈‖‖∇(𝑢_ℎ−𝑢^0ℎ)‖‖₀+𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀+𝑁‖‖∇(𝑢_ℎ−𝑢^0ℎ)‖‖₀‖‖∇𝑢_ℎ‖‖₀+𝐶²⁰𝜆‖‖∇(𝑇_ℎ−𝑇^0ℎ)‖‖₀≤50𝜎ℎ𝐶²⁰𝜆𝜀21(𝜈+𝜎ℎ)+10𝜎ℎ𝐶²⁰𝜆𝜀7+5𝜎ℎ𝐶²⁰𝜆𝜀3𝜈+10𝜎ℎ𝐶²⁰𝜆𝜀21𝜈+2𝜎ℎ𝐶²⁰𝜆𝜀+𝜎ℎ𝐶₂₀7𝜆𝜀+𝜎ℎ𝐶²⁰𝜆+𝜎ℎ𝐶²⁰𝜆≤95𝜎ℎ𝐶₂₀14𝜈𝜆𝜀21(𝜈+𝜎ℎ)+19𝜎ℎ𝐶²⁰𝜆𝜀7+𝜎ℎ𝐶²⁰𝜆+𝜎ℎ𝐶²⁰𝜆14𝜈.(5.20) Hence, we finish the proof.

Theorem 5.5. Under the assumptions of Lemmas 2.3 and 5.3, the following inequality ‖‖∇(𝑢−𝑢^0ℎ)‖‖₀+‖‖𝑝−𝑝^0ℎ‖‖₀+‖‖∇(𝑇−𝑇^0ℎ)‖‖₀≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ,(5.21) holds, where 𝐶 is a positive constant numbers.

Proof. By Lemmas 5.3, 5.4, and the triangle inequality this theorem is obviously true.

Lemma 5.6. For all 𝑢∈𝐻²(Ω)∩𝑋, 𝜔∈𝑊₀, 𝜓∈𝐻²(Ω)∩𝑊₀, there hold that ||||||𝑏𝑢−𝑅_ℎ,𝜔,𝜓|||≤𝐶‖‖𝑢−𝑅_ℎ‖‖₀‖𝒜𝜔‖₀‖∇𝜓‖₀,(5.22)𝑏𝑢,𝑇−𝑟_ℎ̃𝑇,𝜓|||≤𝐶‖𝒜𝑢‖₀‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀‖∇𝜓‖₀.(5.23)

Proof. Letting 𝜔=(𝜔,0)^𝑇, we have 𝑏𝑢−𝑅_ℎ,𝜔,𝜓=𝑏𝑢−𝑅_ℎ,𝜔,𝜓.(5.24) Using (2.25), we can deduce (5.22). Because 𝑇−𝑟_ℎ̃𝑇∈𝑊₀, (5.23) holds.

Theorem 5.7. Under the assumptions of Lemmas 2.3 and 5.3, the following inequality: ‖‖𝑢−𝑢^0ℎ‖‖₀+‖‖𝑇−𝑇^0ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ,(5.25) holds, where 𝐶 is a positive constant.

Proof. Subtracting (3.1) from (2.4) we get the error equations, namely, 𝜈𝑎𝑢−+𝑢𝑑^0ℎ,𝑣𝜑_ℎℎ−𝜎ℎ𝑎𝑢^0ℎ,𝑣_ℎ+𝑏𝑢−𝑢^0ℎ,𝑢^0ℎ,𝑣_ℎ+𝑏𝑢,𝑢−𝑢^0ℎ,𝑣_ℎ−𝑑𝑝−𝑝^0ℎ,𝑣_ℎ,𝑢−𝑢^0ℎ=𝜆𝑗𝑇−𝑇^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝑇−𝑇^0ℎ,𝜓_ℎ−𝜎ℎ𝑎𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢−𝑢^0ℎ,𝑇,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝑇−𝑇^0ℎ,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ.(5.26) Letting 𝑒^0ℎ=𝑅_ℎ−𝑢^0ℎ, 𝜂^0ℎ=𝑄_ℎ−𝑝^0ℎ, 𝜉^0ℎ=𝑟_ℎ̃𝑇−𝑇^0ℎ and using (5.1) and (5.3), we can get 𝜈𝑎𝑒=^0ℎ,𝜆𝑣_ℎ𝑗−𝑇𝜎−ℎ𝑎𝑇_0ℎ𝑢^0ℎ,𝑣_ℎ+𝑏𝑢−𝑢^0ℎ,𝑢^0ℎ,𝑣_ℎ+𝑏𝑢,𝑢−𝑢^0ℎ,𝑣_ℎ−𝑑𝜂^0ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑒^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝜉_ℎ,𝜓_ℎ−𝜎ℎ𝑎𝑇^0ℎ,𝜓_ℎ+𝜆𝑏𝑢−𝑢^0ℎ,𝑇,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝑇−𝑇^0ℎ,𝜓_ℎ=0,∀𝜓_ℎ∈𝑊_0ℎ.(5.27) Taking 𝑣_ℎ=𝑒^0ℎ, 𝜑_ℎ=𝜂^0ℎ in the first equation of (5.27), we obtain 𝜈𝑎𝑒=^0ℎ,𝜆𝑒^0ℎ𝑗−𝑇𝜎−ℎ𝑎𝑇_0ℎ𝑢^0ℎ,𝑒^0ℎ+𝑏𝑒^0ℎ,𝑢^0ℎ,𝑒^0ℎ+𝑏𝑢−𝑅_ℎ,𝑢^0ℎ,𝑒^0ℎ+𝑏𝑢,𝑢−𝑅_ℎ,𝑒^0ℎ,𝑒^0ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ.(5.28) Using (2.10) and (A₁), we deduce 𝜈−𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖²⁰≤+||𝑏||𝜆𝑢𝑗−𝑅𝑇_ℎ−,𝑢𝑇,_0ℎ𝑒^0ℎ||+||𝑏𝑢^0ℎ,𝑢−𝑅_ℎ,𝑒^0ℎ||≤𝑁‖∇𝑢‖₀,𝑒^0ℎ||+||𝜎ℎ𝑎𝑢^0ℎ,𝑒^0ℎ||+‖‖∇𝑢^0ℎ‖‖₀‖‖∇(𝑢−𝑅_ℎ)‖‖₀‖‖∇𝑒^0ℎ‖‖₀+𝐶²⁰𝜆‖‖∇(𝑇−𝑇^0ℎ)‖‖₀‖‖∇𝑒^0ℎ‖‖₀+𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖₀.(5.29) Using Theorem 2.1, (2.16), (4.1), and (5.2), we can obtain ‖‖∇𝑒^0ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ.(5.30) Taking 𝜓_ℎ=𝜉^0ℎ in the second equation of (5.27) and using (2.9) we have 𝑎𝜉^0ℎ,𝜉^0ℎ−𝜎ℎ𝑎𝑇^0ℎ,𝜉^0ℎ+𝜆𝑏𝑢^0ℎ,𝑇−𝑟_ℎ̃𝑇,𝜉^0ℎ+𝜆𝑏𝑢−𝑅_ℎ,𝑇,𝜉^0ℎ+𝜆𝑏𝑒^0ℎ,𝑇,𝜉^0ℎ=0.(5.31) By (2.9), we have 𝑏𝑢^0ℎ=,𝑇−−𝑟_ℎ̃𝑇,𝜉^0ℎ+𝑏𝑢−𝑅_ℎ,𝑇,𝜉^0ℎ+𝑏𝑒^0ℎ,𝑇,𝜉^0ℎ+𝑏𝑒^0ℎ,𝑇−𝑟_ℎ̃𝑇,𝜉^0ℎ−𝑏𝑢−𝑅_ℎ,𝑇−𝑟_ℎ̃𝑇,𝜉^0ℎ+𝑏𝑢,𝑇−𝑟_ℎ̃𝑇,𝜉^0ℎ𝑏𝑢−𝑅_ℎ,𝑇,𝜉^0ℎ+𝑏𝑒^0ℎ,𝑇−𝑟_ℎ̃𝑇,𝜉^0ℎ+𝑏𝑒^0ℎ,𝜉^0ℎ,𝜉^0ℎ+𝑏𝑒^0ℎ,𝑇^0ℎ,𝜉^0ℎ.(5.32) Letting 𝑇=𝜔+𝑇₀, 𝜔∈𝑊₀ and using Lemma 5.6, we can get |||𝑏𝑢≤^0ℎ,𝑇−𝑟_ℎ̃𝑇,𝜉^0ℎ+𝑏𝑢−𝑅_ℎ,𝑇,𝜉^0ℎ+𝑏𝑒_ℎ,𝑇,𝜉^0ℎ|||𝑁‖‖∇𝑒^0ℎ‖‖₀‖‖∇(𝑇−𝑟_ℎ̃𝑇)‖‖₀‖‖∇𝜉^0ℎ‖‖₀+𝑁‖‖∇(𝑢−𝑅_ℎ)‖‖₀‖‖∇(𝑇−𝑟_ℎ̃𝑇)‖‖₀‖‖∇𝜉^0ℎ‖‖₀+𝑁‖‖∇𝑒^0ℎ‖‖₀‖‖∇(𝑇−𝑟_ℎ̃𝑇)‖‖₀‖‖∇𝜉^0ℎ‖‖₀+𝑁‖‖∇𝑒^0ℎ‖‖₀‖‖∇𝑇^0ℎ‖‖₀‖‖∇𝜉^0ℎ‖‖₀+𝐶‖𝒜𝑢‖₀‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀‖‖∇𝜉^0ℎ‖‖₀+𝐶‖‖𝑢−𝑅_ℎ‖‖₀‖𝒜𝜔‖₀‖‖∇𝜉^0ℎ‖‖₀+𝐶‖‖∇(𝑢−𝑅_ℎ)‖‖₀‖‖∇𝑇₀‖‖₀‖‖∇𝜉^0ℎ‖‖₀.(5.33) By assumption (A₂), letting 𝜀<ℎ and using Lemma 5.1 and (2.16), (4.26), and (5.33), we can deduce ‖‖∇𝜉^0ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ.(5.34) Hence, we have ‖‖𝑇−𝑇^0ℎ‖‖₀≤‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀+‖‖𝜉^0ℎ‖‖₀≤‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀+𝐶₀‖‖∇𝜉^0ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ.(5.35)
By (2.10) and (2.25), we can deduce||𝑏𝑢≤−||𝑅𝑏_ℎ,𝑢𝑢−,𝑒𝑅^0ℎ_ℎ||+||𝑏𝑢^0ℎ,𝑢−𝑅_ℎ,𝑒^0ℎ||+||𝑏𝑢−𝑅,_ℎ𝑢,𝑒^0ℎ||+||𝑏𝑢,𝑢−𝑅_ℎ,𝑒^0ℎ||≤𝐶‖𝒜𝑢‖₀,𝑢−𝑅_ℎ,𝑒^0ℎ||+||𝑏𝑒_ℎ,𝑢−𝑅_ℎ,𝑒^0ℎ||‖‖𝑢−𝑅_ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖₀+𝑁‖‖∇𝑢−𝑅_ℎ‖‖₀+‖‖∇𝑒^0ℎ‖‖₀‖‖∇𝑢−𝑅_ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖₀≤𝐶ℎ²‖‖∇𝑒^0ℎ‖‖₀.(5.36) Using (5.29), we can obtain 𝜈−𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖²⁰≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1‖‖∇𝑒^0ℎ‖‖₀+𝐶₀𝜆‖‖𝑇−𝑇^0ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖₀+𝜎ℎ‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑒^0ℎ‖‖₀.(5.37) By using (2.16) and (4.1), there holds 𝜈−𝑁‖‖∇𝑢^0ℎ‖‖₀≥4𝜈5.(5.38) Hence, we can deduce from (5.37) ‖‖∇𝑒^0ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ.(5.39) Therefore, we can deduce ‖𝑢−𝑢^0ℎ‖₀≤‖‖𝑢−𝑅_ℎ‖‖₀+‖‖𝑒^0ℎ‖‖₀≤‖‖𝑢−𝑅_ℎ‖‖₀+𝐶₀‖‖∇𝑒^0ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ.(5.40)

Theorem 5.8. Under the assumptions of Lemmas 2.3 and 5.3, then there holds ‖‖𝑢−‖‖𝑢∇_1ℎ(𝑢−𝑢^1ℎ)‖‖₀+‖‖∇(𝑇−𝑇^1ℎ)‖‖₀≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ²,‖‖₀+‖‖𝑇−𝑇^1ℎ‖‖₀+ℎ‖‖𝑝−𝑝^1ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ²,(5.41) where 𝐶 is a positive constant.

Proof. Subtracting (3.4) from (2.4) we get the error equations, namely, 𝜈𝑎𝑢−−𝑢𝑏^1ℎ,𝑣𝑢_ℎ0ℎ−𝜎ℎ𝑎𝑢^1ℎ,𝑣_ℎ+𝑏𝑢,𝑢,𝑣_ℎ−𝑏𝑢^1ℎ,𝑢^0ℎ,𝑣_ℎ=𝜆𝑗𝑇,−𝑢^1ℎ𝑇,𝑣_ℎ1ℎ−𝑑𝑝−𝑝^1ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑢−𝑢^1ℎ,𝑣_ℎ−𝑏𝑢^0ℎ,𝑢^0ℎ,𝑣_ℎ−𝜎ℎ𝑎𝑢^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,𝑎𝑇=−𝑇−^1ℎ𝜎,ℎ𝜓_ℎ−𝜎ℎ𝑎𝑇^1ℎ,𝜓_ℎ+𝜆𝑏𝑢,𝑇,𝜓_ℎ−𝜆𝑏𝑢^1ℎ,𝑇^0ℎ,𝜓_ℎ−𝜆𝑏𝑢^0ℎ,𝑇^1ℎ,𝜓_ℎ𝑎𝑇^0ℎ,𝜓_ℎ−𝜆𝑏𝑢^0ℎ,𝑇^0ℎ,𝜓_ℎ,∀𝜓_ℎ∈𝑊_0ℎ.(5.42)
Letting 𝑒^1ℎ=𝑅_ℎ−𝑢^1ℎ, 𝜂^1ℎ=𝑄_ℎ−𝑝^1ℎ, 𝜉^1ℎ=𝑟_ℎ̃𝑇−𝑇^1ℎ, using (5.1) and (5.3) and adding and subtracting appropriate terms in the above expression yields(𝜈+=𝜎ℎ𝜆)𝑎𝑗𝑒𝑇^1ℎ,−𝑣_ℎ𝑇_0ℎ+𝑏𝑢^0ℎ,𝑢−𝑢^1ℎ,𝑣_ℎ+𝑏𝑢−𝑢^1ℎ,𝑢^0ℎ,𝑣_ℎ−𝑑𝜂^1ℎ,𝑣_ℎ+𝑑𝜑_ℎ,𝑒^1ℎ(1+𝜎ℎ),𝑣_ℎ+𝜎ℎ𝑎𝑅_ℎ−𝑢^0ℎ,𝑣_ℎ−𝑏𝑢−𝑢^0ℎ,𝑢−𝑢^0ℎ,𝑣_ℎ,∀𝑣_ℎ∈𝑋_ℎ,𝜑_ℎ∈𝑀_ℎ,=𝜎ℎ𝑎𝜉^1ℎ,𝜓_ℎ+𝜆𝑏𝑢^0ℎ,𝑇−𝑇^1ℎ,𝜓_ℎ−𝜆𝑏𝑢−𝑢^1ℎ,𝑇^0ℎ,𝜓_ℎ𝑎𝑟_ℎ̃𝑇−𝑇^0ℎ,𝜓_ℎ−𝜆𝑏𝑢−𝑢^0ℎ,𝑇−𝑇^0ℎ,𝜓_ℎ,∀𝜓_ℎ∈𝑊_0ℎ.(5.43) Letting 𝑣_ℎ=𝑒^1ℎ, 𝜑_ℎ=𝜂^1ℎ in the first equation of (5.43), we can deduce (𝜈+=𝜎ℎ𝜆)𝑎𝑗𝑒𝑇^1ℎ,−𝑒^1ℎ𝑇_0ℎ+𝑏𝑢^0ℎ,𝑢−𝑅_ℎ,𝑒^1ℎ+𝑏𝑢−𝑅_ℎ,𝑢^0ℎ,𝑒^1ℎ+𝑏𝑒^1ℎ,𝑢^0ℎ,𝑒^1ℎ,𝑒^1ℎ+𝜎ℎ𝑎𝑅_ℎ−𝑢^0ℎ,𝑒^1ℎ−𝑏𝑢−𝑢^0ℎ,𝑢−𝑢^0ℎ,𝑒^1ℎ(5.44) By (2.10) and (2.25), we can deduce 𝜈+𝜎ℎ−𝑁‖‖∇𝑢^0ℎ‖‖₀‖‖∇𝑒^1ℎ‖‖₀≤𝐶‖‖𝒜𝑢^0ℎ‖‖₀‖‖𝑢−𝑅_ℎ‖‖₀+𝜎ℎ‖‖∇(𝑅_ℎ−𝑢^0ℎ)‖‖₀+𝑁‖‖∇(𝑢−𝑢^0ℎ)‖‖²⁰+𝜆𝐶₀‖‖𝑇−𝑇^0ℎ‖‖₀.(5.45) Using (2.16), (4.1), (4.26), (5.21), and (5.2), we can obtain ‖‖∇𝑒^1ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ².(5.46) Using (5.2) and triangle inequality, we can have ‖‖∇(𝑢−𝑢^1ℎ)‖‖₀≤‖‖∇(𝑢−𝑅_ℎ)‖‖₀+‖‖∇𝑒₁‖‖₀‖‖𝑢−𝑢_1ℎ≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ²,‖‖₀≤‖‖𝑢−𝑅_ℎ‖‖₀+‖‖𝑒₁‖‖₀≤‖‖𝑢−𝑅_ℎ‖‖₀+𝐶₀‖‖∇𝑒₁‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ².(5.47)
Letting 𝜓_ℎ=𝜉^1ℎ in the second equation of (5.43) and using (2.8), we can deduce(1+=𝜎ℎ𝜎)ℎ𝑎𝜉^1ℎ,𝜉^1ℎ+𝜆𝑏𝑢^0ℎ,𝑇−𝑟_ℎ̃𝑇,𝜉^1ℎ−𝜆𝑏𝑢−𝑢^1ℎ,𝑇^0ℎ,𝜉^1ℎ𝑎𝑇−𝑇^0ℎ,𝜉^1ℎ−𝜆𝑏𝑢−𝑢^0ℎ,𝑇−𝑇^0ℎ,𝜉^1ℎ.(5.48) Letting 𝑇^0ℎ=𝜔^0ℎ+𝑇₀ and using (2.11), (5.22), and (5.23), we have (1+𝜎ℎ)‖‖∇𝜉^1ℎ‖‖₀≤𝐶𝜆‖‖𝒜𝑢^0ℎ‖‖₀‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀+𝐶𝜆‖‖𝑢−𝑢^1ℎ‖‖₀‖‖𝒜𝜔^0ℎ‖‖₀+𝑁𝜆‖‖∇(𝑢−𝑢^1ℎ)‖‖₀‖‖∇𝑇₀‖‖₀+𝜎ℎ‖‖∇(𝑇−𝑇^0ℎ)‖‖₀+𝜆𝑁‖‖∇(𝑢−𝑢^0ℎ)‖‖₀‖‖∇(𝑇−𝑇^0ℎ)‖‖₀.(5.49) Using (5.5), (5.21), (5.47), we can obtain ‖‖∇𝜉^1ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ².(5.50) Using (5.2) and triangle inequality, we can have ‖‖∇(𝑇−𝑇^1ℎ)‖‖₀≤‖‖∇(𝑇−𝑟_ℎ̃𝑇)‖‖₀+‖‖∇𝜉^1ℎ‖‖₀‖‖𝑇−𝑇_1ℎ≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ²,‖‖₀≤‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀+‖‖𝜉^1ℎ‖‖₀≤‖‖𝑇−𝑟_ℎ̃𝑇‖‖₀+𝐶₀‖‖∇𝜉^1ℎ‖‖₀≤𝐶ℎ^𝑟+1‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ².(5.51)
Taking 𝜑_ℎ=0, 𝑣_ℎ=𝑅_ℎ−𝑢^1ℎ in the first equation of (5.43) and using (2.3), we have𝛽‖‖𝜂^1ℎ‖‖₀≤(𝜈+𝜎ℎ)‖‖∇𝑒^1ℎ‖‖₀+𝜎ℎ‖‖∇(𝑢−𝑢^0ℎ)‖‖₀+2𝑁‖‖∇𝑢^0ℎ‖‖‖‖∇(𝑢−𝑢^1ℎ)‖‖₀+𝐶₀𝜆‖‖𝑇−𝑇^0ℎ‖‖₀+𝑁‖‖∇(𝑢−𝑢^0ℎ)‖‖²⁰.(5.52) By (4.1), (5.21), and (5.47), we can deduce ‖‖𝜂^1ℎ‖‖₀≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ².(5.53) Using (5.2) and triangle inequality, we can have ‖‖𝑝−𝑝^1ℎ‖‖₀≤‖‖𝑝−𝑄_ℎ‖‖₀+‖‖𝜂^1ℎ‖‖₀≤𝐶ℎ^𝑟‖𝑢‖_𝑟+1+‖𝑝‖_𝑟+‖𝑇‖_𝑟+1+𝐶ℎ².(5.54)

6. Numerical Experiments

In this section, we present some numerical examples with a physical model of square cavity stationary flow. We choose different 𝜈 for comparison. The side length of the square cavity and the boundary conditions are given by Figure 1. From Figure 1, we can see that the 𝑇=0 on left and lower boundaries, 𝜕𝑇/𝜕𝑛=0 on upper boundary, and 𝑇=4𝑦(1−𝑦) on right boundary of the cavity. We use 𝑃₂−𝑃₁−𝑃₂ finite element here.

Figure 1 

Physics model of the cavity flows.

Firstly, we choose 𝜈=1/2000, 𝜎=0.4 and divide the cavity into 𝑀×𝑁=40×40, that is, ℎ=√2/40. Figure 2 gives the numerical isotherms (a) and the numerical isobar (b). Figure 3 gives the numerical streamline. From the numerical results, we can see that our method is stable and has a good precision.

(a)

(b)

Figure 2 

The numerical Isotherms (a) and the numerical Isobar (b) for 𝜈=1/2000 by the defect-correction MFEM with ℎ=√2/40, 𝜎=0.4.

Figure 3 

The numerical streamline for 𝜈=1/2000 by the defect-correction MFEM with ℎ=√2/40, 𝜎=0.4.

Secondly, we choose 𝜈=1/5000, 𝜎=0.4 to show our our method suiting for solving the conduction convection problems with small viscosity. It is well known that it is more and more difficult to solve the problem by numerical method as 𝜈 changing smaller and smaller. Hence, we divide the cavity into 𝑀×𝑁=100×100, namely ℎ=√2/100. Figure 4 gives the numerical isotherms (a) and the numerical isobar (b), and Figure 5 shows the numerical streamline. At last, we choose 𝜈=1/6000, 𝜎=0.4. Figure 6 gives the numerical isotherms (a) and the numerical isobar (b), and Figure 7 shows the numerical streamline.

(a)

(b)

Figure 4 

The numerical Isotherms (a) and the numerical Isobar (b) for 𝜈=1/5000 by the defect-correction MFEM with ℎ=√2/100, 𝜎=0.4.

Figure 5 

The numerical streamline for 𝜈=1/5000 by the defect-correction MFEM with ℎ=√2/100, 𝜎=0.4.

(a)

(b)

Figure 6 

The numerical Isotherms (a) and the numerical Isobar (b) for 𝜈=1/6000 by the defect-correction MFEM with ℎ=√2/100, 𝜎=0.4.

Figure 7 

The numerical streamline for 𝜈=1/6000 by the defect-correction MFEM with ℎ=√2/100, 𝜎=0.4.

Just as Remark 3.1, we only use one correction step in our numerical experiments. From the numerical, we can see that when 𝜈=0.5×10⁻³ the numerical streamline is very regular. The pressure is small near the wall. But the numerical streamline changes more and more immethodical with 𝜈 changing smaller and smaller. And the pressure changes bigger near the wall. In conclusion, the defect-correction MFEM is highly efficient for the stationary conduction-convection problems and it can be used for solving the convection-conduction problems with much small viscosity.

Acknowledgments

The authors would like to thank the editor and the referees for their criticism, valuable comments, which led to the improvement of this paper. This work is supported by the NSF of China (no. 10971166) and the National High Technology Research and Development program of China (863 program, no. 2009AA01A135).