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Mathematical Problems in Engineering
VolumeΒ 2011Β (2011), Article IDΒ 370192, 28 pages
http://dx.doi.org/10.1155/2011/370192
Research Article

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

Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, China

Received 29 July 2010; Revised 15 November 2010; Accepted 5 January 2011

Academic Editor: Katica R. (Stevanovic)Β Hedrih

Copyright Β© 2011 Zhiyong Si and Yinnian He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑒, 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 (𝑒,𝑝,𝑇)βˆˆπ‘‹Γ—π‘€Γ—π‘Š such thatβˆ’πœˆΞ”π‘’+(π‘’β‹…βˆ‡)𝑒+βˆ‡π‘=πœ†π‘—π‘‡,π‘₯∈Ω,div𝑒=0,π‘₯∈Ω,βˆ’Ξ”π‘‡+πœ†π‘’β‹…βˆ‡π‘‡=0,π‘₯∈Ω,𝑒=0,𝑇=𝑇0,π‘₯βˆˆπœ•Ξ©,(1.1) where Ξ© is a bounded domain in ℝ2 assumed to have a Lipschitz continuous boundary πœ•Ξ©. 𝑒=(𝑒1(π‘₯),𝑒2(π‘₯))𝑇 represents the velocity vector, 𝑝(π‘₯) the pressure, 𝑇(π‘₯) the temperature, πœ†>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 𝑋=𝐻10(Ξ©)2,π‘Š=𝐻1(Ξ©),𝑀=𝐿20ξ‚»(Ξ©)β‰πœ‘βˆˆπΏ2ξ€œ(Ξ©);Ξ©ξ‚Ό.πœ‘π‘‘π‘₯=0(2.1)β„‘β„Ž is the uniformly regular family of triangulation of Ξ©, indexed by a parameter β„Ž=maxπΎβˆˆβ„‘β„Ž{β„ŽπΎ;β„ŽπΎ=diam(𝐾)}. We introduce the finite element subspace π‘‹β„ŽβŠ‚π‘‹, π‘€β„ŽβŠ‚π‘€, π‘Šβ„ŽβŠ‚π‘Š as follows π‘‹β„Ž=ξ‚»π‘£β„Žβˆˆπ‘‹βˆ©πΆ0Ω2;π‘£β„Ž|πΎβˆˆπ‘ƒβ„“(𝐾)2,βˆ€πΎβˆˆβ„‘β„Žξ‚Ό,π‘€β„Ž=ξ‚†π‘žβ„Žβˆˆπ‘€βˆ©πΆ0Ω;π‘žβ„Ž|πΎβˆˆπ‘ƒπ‘˜(𝐾),βˆ€πΎβˆˆβ„‘β„Žξ‚‡,π‘Šβ„Ž=ξ‚†πœ™β„Žβˆˆπ‘Šβˆ©πΆ0Ω;πœ™β„Ž|πΎβˆˆπ‘ƒπ‘™(𝐾),βˆ€πΎβˆˆβ„‘β„Žξ‚‡,(2.2) where 𝑃ℓ(𝐾) is the space of piecewise polynomials of degree β„“ on 𝐾, and β„“β©Ύ1, π‘˜β©Ύ1, 𝑙⩾1 are three integers. π‘Š0β„Ž=π‘Šβ„Žβˆ©π»10(Ξ©), and (π‘‹β„Ž,π‘€β„Ž) satisfies the discrete LBB conditionsupπ‘£β„Žβˆˆπ‘‹β„Žπ‘‘ξ€·πœ‘β„Ž,π‘£β„Žξ€Έβ€–β€–βˆ‡π‘£β„Žβ€–β€–0β€–β€–πœ‘β©Ύπ›½β„Žβ€–β€–0,βˆ€πœ‘β„Žβˆˆπ‘€β„Ž,(2.3) where 𝑑(πœ‘,𝑣)=(πœ‘,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.

(π’«ξ…ž) Find (𝑒,𝑝,𝑇)βˆˆπ‘‹Γ—π‘€Γ—π‘Š such thatπœˆπ‘Ž(𝑒,𝑣)βˆ’π‘‘(𝑝,𝑣)+𝑑(πœ‘,𝑒)+𝑏(𝑒,𝑒,𝑣)=πœ†(𝑗𝑇,𝑣),βˆ€π‘£βˆˆπ‘‹,πœ‘βˆˆπ‘€,π‘Ž(𝑇,πœ“)+πœ†π‘(𝑒,𝑇,πœ“)=0,βˆ€πœ“βˆˆπ‘Š0.(2.4)

(π’«ξ…žξ…ž) Find (π‘’β„Ž,π‘β„Ž,π‘‡β„Ž)βˆˆπ‘‹β„ŽΓ—π‘€β„ŽΓ—π‘Šβ„Ž such thatξ€·π‘’πœˆπ‘Žβ„Ž,π‘£β„Žξ€Έξ€·π‘βˆ’π‘‘β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,π‘’β„Žξ€Έξ€·π‘’+π‘β„Ž,π‘’β„Ž,π‘£β„Žξ€Έξ€·=πœ†π‘—π‘‡β„Ž,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,π‘Žξ€·π‘‡β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Ž,π‘‡β„Ž,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž,(2.5) where π‘Ž(𝑒,𝑣)=(βˆ‡π‘’,βˆ‡π‘£), 𝑑(πœ‘,𝑣)=(πœ‘,div𝑣), π‘Ž(𝑇,πœ“)=(βˆ‡π‘‡,βˆ‡πœ“), and 1𝑏(𝑒,𝑣,𝑀)=2ξƒ¬ξ€œΞ©2𝑖,π‘˜=1π‘’π‘–πœ•π‘£π‘˜πœ•π‘₯π‘–π‘€π‘˜π‘‘π‘₯βˆ’2𝑖,π‘˜=1π‘’π‘–πœ•π‘€π‘˜πœ•π‘₯π‘–π‘£π‘˜ξƒ­π‘‘π‘₯,βˆ€π‘’,𝑣,π‘€βˆˆπ‘‹,1𝑏(𝑒,𝑇,πœ“)=2ξƒ¬ξ€œΞ©2𝑖=1π‘’π‘–πœ•π‘‡πœ•π‘₯π‘–πœ“π‘‘π‘₯βˆ’2ξ“π‘–π‘’π‘–πœ•πœ“πœ•π‘₯𝑖𝑇𝑑π‘₯,βˆ€π‘’βˆˆπ‘‹,𝑇,πœ“βˆˆπ‘Š.(2.6)

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

(A1)There exists a constant 𝐢0 which only depends on Ξ©, such that(i)‖𝑒‖0≀𝐢0β€–βˆ‡π‘’β€–0, ‖𝑒‖0,4≀𝐢0β€–βˆ‡π‘’β€–0, for all π‘’βˆˆπ»10(Ξ©)2(or𝐻10(Ξ©)),(ii)‖𝑒‖0,4≀𝐢0‖𝑒‖1,for all π‘’βˆˆπ»1(Ξ©)2,(iii)‖𝑒‖0,4β‰€βˆš2β€–βˆ‡π‘’β€–01/2‖𝑒‖01/2,for allπ‘’βˆˆπ»10(Ξ©)2(or𝐻10(Ξ©)).(A2)Assuming πœ•Ξ©βˆˆπΆπ‘˜,𝛼(π‘˜β©Ύ0,𝛼>0), then, for 𝑇0βˆˆπΆπ‘˜,𝛼(πœ•Ξ©), there exists an extension in 𝐢0π‘˜,𝛼(ℝ2) (denote 𝑇0 also), such that ‖‖𝑇0β€–β€–π‘˜,π‘žβ‰€πœ€,π‘˜β©Ύ0,1β‰€π‘žβ‰€βˆž,(2.7) where πœ€ is an arbitrary positive constant.(A3)𝑏(β‹…,β‹…,β‹…) and 𝑏(β‹…,β‹…,β‹…) have the following properties.(i)For all π‘’βˆˆπ‘‹, 𝑣,π‘€βˆˆπ‘‹(or𝑇,πœ“βˆˆπ»10(Ξ©)), there holds that 𝑏(𝑒,𝑣,𝑣)=0,𝑏(𝑒,𝑣,𝑀)=βˆ’π‘(𝑒,𝑀,𝑣),(2.8)𝑏(𝑒,𝑇,𝑇)=0,𝑏(𝑒,𝑇,πœ“)=βˆ’π‘(𝑒,πœ“,𝑇).(2.9)(ii)For allπ‘’βˆˆπ‘‹, π‘£βˆˆπ»1(Ξ©)2(orπ‘‡βˆˆπ»1(Ξ©)), for allπ‘€βˆˆπ‘‹(orπœ“βˆˆπ»10(Ξ©)), there holds that ||||𝑏(𝑒,𝑣,𝑀)β‰€π‘β€–βˆ‡π‘’β€–0β€–βˆ‡π‘£β€–0β€–βˆ‡π‘€β€–0|||,(2.10)|||≀𝑏(𝑒,𝑇,πœ“)π‘β€–βˆ‡π‘’β€–0β€–βˆ‡π‘‡β€–0β€–βˆ‡πœ“β€–0,(2.11) where 𝑁=sup𝑒,𝑣,𝑀||||𝑏(𝑒,𝑣,𝑀)ξ€·β€–βˆ‡π‘’β€–0β€–βˆ‡π‘£β€–0β€–βˆ‡π‘€β€–0ξ€Έ,𝑁=sup𝑒,𝑇,πœ‘||||||𝑏(𝑒,𝑇,πœ‘)ξ€·β€–βˆ‡π‘’β€–0β€–βˆ‡π‘‡β€–0β€–βˆ‡πœ‘β€–0ξ€Έ.(2.12)

We recall the following existence, uniqueness and regularity result of (π’«ξ…ž) (see [7, Chapter 4]).

Theorem 2.1 (see [7]). Under the assumption of (A1)~(A3), letting 𝐴≑2πœˆβˆ’1πœ†(3𝐢0+1)‖𝑇0β€–1, 𝐡≑2β€–βˆ‡π‘‡0β€–0+2(𝐢20πœ†)βˆ’1𝐴, there exist 0<𝛿1, 𝛿2≀1 such that πœˆβˆ’1𝑁𝐴≀1βˆ’π›Ώ1,𝛿1βˆ’1πœˆβˆ’1𝐢20πœ†2𝐡𝑁≀1βˆ’π›Ώ2.(2.13) Then, there exists a unique solution (𝑒,𝑝,𝑇)βˆˆπ‘‹Γ—π‘€Γ—π‘Š for (π’«ξ…ž), and β€–βˆ‡π‘’β€–0≀𝐴,β€–βˆ‡π‘‡β€–0≀𝐡.(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: ||π‘ξ€·π‘’β„Ž,π‘£β„Žξ€Έ||+||𝑏𝑣,π‘€β„Ž,π‘’β„Žξ€Έ||+||𝑏,𝑀𝑀,π‘’β„Ž,π‘£β„Žξ€Έ||≀𝐢0||||logβ„Ž1/2β€–β€–βˆ‡π‘£β„Žβ€–β€–0β€–β€–βˆ‡π‘’β„Žβ€–β€–0‖𝑀‖0,(2.15) for all π‘’β„Ž,π‘£β„Žβˆˆπ‘‰β„Ž, π‘€βˆˆπΏ2(Ξ©)2.

Lemma 2.3. Suppose that (A1)~(A3) are valid and πœ€ is a positive constant, such that 32𝐢20πœ†2π‘πœ€β€–β€–3𝜈<1,βˆ‡π‘‡0β€–β€–0β‰€πœ€4,‖‖𝑇0β€–β€–0≀𝐢0πœ€4,(2.16) then (π’«ξ…žξ…ž) has a unique solution (π‘’β„Ž,π‘β„Ž,π‘‡β„Ž)βˆˆπ‘‹β„ŽΓ—π‘€β„ŽΓ—π‘Šβ„Ž, such that 𝑇|πœ•Ξ©=𝑇0 and β€–β€–βˆ‡π‘’β„Žβ€–β€–0≀5𝐢20πœ†πœ€,β€–β€–3πœˆβˆ‡π‘‡β„Žβ€–β€–0β‰€πœ€.(2.17)

Proof. The proof of the existence and the uniqueness of the solution has been given by Luo [7]. Let π‘‡β„Ž=πœ”β„Ž+𝑇0, πœ“β„Ž=πœ”β„Ž in (2.5), we can get π‘Žξ€·πœ”β„Ž,πœ”β„Žξ€Έ=βˆ’πœ†π‘ξ€·π‘’β„Ž,𝑇0,πœ”β„Žξ€Έβˆ’π‘Žξ€·π‘‡0,πœ”β„Žξ€Έ.(2.18) Using (2.11) and (2.16), we deduce β€–β€–βˆ‡πœ”β„Žβ€–β€–0β‰€β€–β€–βˆ‡π‘‡0β€–β€–0+πœ†β€–β€–π‘πœ€βˆ‡π‘’β„Žβ€–β€–0.(2.19) Letting π‘£β„Ž=π‘’β„Ž, πœ‘β„Ž=π‘β„Ž in the first equation of (2.5), we get πœˆβ€–β€–βˆ‡π‘’β„Žβ€–β€–20=||πœ†ξ€·π‘—π‘‡β„Ž,π‘’β„Žξ€Έ||β‰€πœ†πΆ0β€–β€–π‘‡β„Žβ€–β€–0β€–β€–βˆ‡π‘’β„Žβ€–β€–0.(2.20) By (2.16), we can obtian β€–β€–βˆ‡π‘’β„Žβ€–β€–0β‰€πœˆβˆ’1πœ†πΆ0β€–β€–π‘‡β„Žβ€–β€–0β‰€πœˆβˆ’1πœ†πΆ0ξ€·β€–β€–πœ”β„Žβ€–β€–0+‖‖𝑇0β€–β€–0ξ€Έβ‰€πœˆβˆ’1πœ†πΆ20β€–β€–βˆ‡πœ”β„Žβ€–β€–0+πœˆβˆ’1πœ†πΆ0‖‖𝑇0β€–β€–0β‰€πœˆβˆ’1πœ†πΆ0‖‖𝑇0β€–β€–0+πœˆβˆ’1πœ†πΆ20β€–β€–βˆ‡π‘‡0β€–β€–0+πœˆβˆ’1πœ†2𝐢20β€–β€–π‘πœ€βˆ‡π‘’β„Žβ€–β€–0.(2.21) Using (2.16) again, we get β€–β€–βˆ‡π‘’β„Žβ€–β€–0≀5𝐢20πœ†πœ€.3𝜈(2.22) By (2.19), we deduce β€–β€–βˆ‡π‘‡β„Žβ€–β€–0β‰€β€–β€–βˆ‡πœ”β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0‖‖≀2βˆ‡π‘‡0β€–β€–0+πœ†β€–β€–π‘πœ€βˆ‡π‘’β„Žβ€–β€–0‖‖≀2βˆ‡π‘‡0β€–β€–0+5𝐢20πœ†2π‘πœ€2β‰€πœ€3𝜈2+πœ€2=πœ€.(2.23)

We introduce the Laplace operatorπ’œπ‘’=βˆ’Ξ”π‘’,βˆ€π‘’βˆˆπ·(π’œ)=𝐻2(Ξ©)2βˆ©π‘‹.(2.24)

Lemma 2.4 (see [35, 36]). For all 𝑒,π‘€βˆˆπ‘‹, π‘£βˆˆπ·(𝐴) there holds that ||||+||||+||||𝑏(𝑒,𝑣,𝑀)𝑏(𝑣,𝑒,𝑀)𝑏(𝑀,𝑒,𝑣)β‰€πΆβ€–π’œπ‘£β€–0‖𝑀‖0β€–βˆ‡π‘’β€–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 πœŽβ„Ž as a stabilizing factor in the solution algorithm. For a fixed grid parameter β„Ž 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.

(π’«βˆ—) Find (𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Ž)βˆˆπ‘‹β„ŽΓ—π‘€β„ŽΓ—π‘Šβ„Ž such that𝑒(𝜈+πœŽβ„Ž)π‘Ž0β„Ž,π‘£β„Žξ€Έξ€·π‘βˆ’π‘‘0β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,𝑒0β„Žξ€Έξ€·π‘’+𝑏0β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έξ€·=πœ†π‘—π‘‡0β„Ž,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,(1+πœŽβ„Ž)π‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,𝑇0β„Ž,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(3.1) We define the residual or named defect 𝑅(𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Ž), 𝑄(𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Ž) for the momentum systems as follows:𝑅𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Žξ€Έ,π‘£β„Žξ€Έξ€·=πœ†π‘—π‘‡0β„Ž,π‘£β„Žξ€Έξ€·π‘’βˆ’πœˆπ‘Ž0β„Ž,π‘£β„Žξ€Έξ€·π‘+𝑑0β„Ž,π‘£β„Žξ€Έξ€·πœ‘βˆ’π‘‘β„Ž,𝑒0β„Žξ€Έξ€·π‘’βˆ’π‘0β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έ,𝑄𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Žξ€Έ,πœ“β„Žξ€Έ=βˆ’π‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έβˆ’πœ†π‘ξ€·π‘’0β„Ž,𝑇0β„Ž,πœ“β„Žξ€Έ.(3.2) Define the correction (πœ€0β„Ž,𝜚0β„Ž,𝜏0β„Ž) satisfying the following linear problem:ξ€·πœ€(𝜈+πœŽβ„Ž)π‘Ž0β„Ž,π‘£β„Žξ€Έξ€·πœšβˆ’π‘‘0β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,πœ€0β„Žξ€Έξ€·πœ€+𝑏0β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έξ€·π‘’+𝑏0β„Ž,πœ€0β„Ž,π‘£β„Žξ€Έ=𝑅𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Žξ€Έ,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,(1+πœŽβ„Ž)π‘Žξ€·πœ0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,𝜏0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·πœ€0β„Ž,𝑇0β„Ž,πœ“β„Žξ€Έ=𝑄𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Žξ€Έ,πœ“β„Žξ€Έ,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(3.3) Define 𝑒1β„Ž=𝑒0β„Ž+πœ€0, 𝑝1β„Ž=𝑝0β„Ž+𝜚0β„Ž, 𝑇1β„Ž=𝑇0β„Ž+𝜏0β„Ž, which are hoped to be better solutions of the problems. In order to obtain the equations for (𝑒1β„Ž,𝑝1β„Ž,𝑇1β„Ž), we use the residual equation (3.2) to rewrite the linear problems (3.3); we obtainξ€·π’«β€ ξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ€·π‘’(𝜈+πœŽβ„Ž)π‘Ž1β„Ž,π‘£β„Žξ€Έξ€·π‘βˆ’π‘‘1β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,𝑒1β„Žξ€Έξ€·π‘’+𝑏0β„Ž,𝑒1β„Ž,π‘£β„Žξ€Έξ€·π‘’+𝑏1β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έξ€·=πœ†π‘—π‘‡1β„Ž,π‘£β„Žξ€Έξ€·π‘’+πœŽβ„Žπ‘Ž0β„Ž,π‘£β„Žξ€Έξ€·π‘’+𝑏0β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,(1+πœŽβ„Ž)π‘Žξ€·π‘‡1β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’1β„Ž,𝑇0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,𝑇1β„Ž,πœ“β„Žξ€Έ=πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,𝑇0β„Ž,πœ“β„Žξ€Έ,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(3.4) In general, this method can be described as follows.

Step 1. Solve the nonlinear systems (3.1) for (𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Ž).

Step 2. For 𝑖=1,2,…,π‘š, solve the linear equations ξ€·π’«β€‘ξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξ€·π‘’(𝜈+πœŽβ„Ž)π‘Žπ‘–β„Ž,π‘£β„Žξ€Έξ€·π‘βˆ’π‘‘π‘–β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,π‘’π‘–β„Žξ€Έξ€·π‘’+π‘β„Žπ‘–βˆ’1,π‘’π‘–β„Ž,π‘£β„Žξ€Έξ€·π‘’+π‘π‘–β„Ž,π‘’β„Žπ‘–βˆ’1,π‘£β„Žξ€Έ=ξ€·π‘‡π‘–β„Ž,π‘£β„Žξ€Έξ€·π‘’+πœŽβ„Žπ‘Žβ„Žπ‘–βˆ’1,π‘£β„Žξ€Έξ€·π‘’+π‘β„Žπ‘–βˆ’1,π‘’β„Žπ‘–βˆ’1,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,(1+πœŽβ„Ž)π‘Žξ€·π‘‡π‘–β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’π‘–β„Ž,π‘‡β„Žπ‘–βˆ’1,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Žπ‘–βˆ’1,π‘‡π‘–β„Ž,πœ“β„Žξ€Έ=πœŽβ„Žπ‘Žξ€·π‘‡β„Žπ‘–βˆ’1,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Žπ‘–βˆ’1,π‘‡β„Žπ‘–βˆ’1,πœ“β„Žξ€Έ,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(3.5) For each 𝑖 the residual is given by ξ€·π‘…ξ€·π‘’π‘–β„Ž,π‘π‘–β„Ž,π‘‡π‘–β„Žξ€Έ,π‘£β„Žξ€Έξ€·=πœ†π‘—π‘‡π‘–β„Ž,π‘£β„Žξ€Έξ€·π‘’βˆ’πœˆπ‘Žπ‘–β„Ž,π‘£β„Žξ€Έξ€·π‘+π‘‘π‘–β„Ž,π‘£β„Žξ€Έξ€·πœ‘βˆ’π‘‘β„Ž,π‘’π‘–β„Žξ€Έξ€·π‘’βˆ’π‘π‘–β„Ž,π‘’π‘–β„Ž,π‘£β„Žξ€Έ,ξ€·π‘„ξ€·π‘’π‘–β„Ž,π‘π‘–β„Ž,π‘‡π‘–β„Žξ€Έ,πœ“β„Žξ€Έ=βˆ’π‘Žξ€·π‘‡π‘–β„Ž,πœ“β„Žξ€Έβˆ’πœ†π‘ξ€·π‘’π‘–β„Ž,π‘‡π‘–β„Ž,πœ“β„Žξ€Έ.(3.6) The correction (πœ€π‘–β„Ž,πœšπ‘–β„Ž,πœπ‘–β„Ž) is given by ξ€·πœ€(𝜈+πœŽβ„Ž)π‘Žπ‘–β„Ž,π‘£β„Žξ€Έξ€·πœšβˆ’π‘‘π‘–β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,πœ€π‘–β„Žξ€Έξ€·πœ€+π‘π‘–β„Ž,π‘’π‘–β„Ž,π‘£β„Žξ€Έξ€·π‘’+π‘π‘–β„Ž,πœ€π‘–β„Ž,π‘£β„Žξ€Έ=ξ€·π‘…ξ€·π‘’π‘–β„Ž,π‘π‘–β„Ž,π‘‡π‘–β„Žξ€Έ,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,(1+πœŽβ„Ž)π‘Žξ€·πœπ‘–β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’π‘–β„Ž,πœπ‘–,πœ“β„Žξ€Έ+πœ†π‘ξ€·πœ€π‘–β„Ž,π‘‡π‘–β„Ž,πœ“β„Žξ€Έ=ξ€·π‘„ξ€·π‘’π‘–β„Ž,π‘π‘–β„Ž,π‘‡π‘–β„Žξ€Έ,πœ“β„Žξ€Έ,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(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 (𝑒0β„Ž,𝑇0β„Ž) defined by (π’«βˆ—) satisfies β€–β€–βˆ‡π‘’0β„Žβ€–β€–0≀5𝐢20πœ†πœ€,β€–β€–3(𝜈+πœŽβ„Ž)βˆ‡π‘‡0β„Žβ€–β€–0β‰€πœ€.(4.1) Moreover, if 25𝐢20π‘πœ†πœ€3(𝜈+πœŽβ„Ž)2<1,(4.2)(π’«βˆ—) admits a unique solution.

Proof. We define the set ℬ𝑀=ξƒ―Μƒπ‘£β„Žβˆˆπ‘‹β„Ž;β€–β€–βˆ‡Μƒπ‘£β„Žβ€–β€–0≀5𝐢20πœ†πœ€ξƒ°3(𝜈+πœŽβ„Ž).(4.3)
Let Μƒπ‘’β„Ž be in ℬ𝑀. Then(1+πœŽβ„Ž)π‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·Μƒπ‘’β„Ž,𝑇0β„Ž,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž(4.4) has a unique solution 𝑇0β„Žβˆˆπ‘Šβ„Ž such that π‘‡β„Ž|πœ•Ξ©=𝑇0. For a given 𝑇0β„Ž, we consider the following problem: 𝑒(𝜈+πœŽβ„Ž)π‘Žβ„Ž0βˆ—,π‘£β„Žξ€Έξ€·π‘βˆ’π‘‘β„Ž0βˆ—,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,π‘’β„Ž0βˆ—ξ€Έξ€·π‘’+π‘β„Ž0βˆ—,π‘’β„Ž0βˆ—,π‘£β„Žξ€Έξ€·=πœ†π‘—π‘‡0β„Ž,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž.(4.5) By the theory of the Navier-Stokes equations, we get (4.5) has a unique solution (π‘’β„Ž0βˆ—,π‘β„Ž0βˆ—)βˆˆπ‘‹β„ŽΓ—π‘€β„Ž (see [31]). It means that (4.4) and (4.5) give a unique π‘’β„Ž0βˆ—βˆˆπ‘‹β„Ž for a given Μƒπ‘’β„Žβˆˆπ‘‹β„Ž, we denote π‘’β„Ž0βˆ—=β„“β„ŽΜƒπ‘’β„Ž.(4.6)
Setting 𝑇0β„Ž=πœ”0β„Ž+𝑇0, πœ“β„Ž=πœ”0β„Ž in (4.4) and using (2.9), we can obtain(1+πœŽβ„Ž)π‘Žξ€·πœ”0β„Ž,πœ”0β„Žξ€Έ=βˆ’πœ†π‘ξ€·Μƒπ‘’β„Ž,𝑇0,πœ”0β„Žξ€Έβˆ’(1+πœŽβ„Ž)π‘Žξ€·π‘‡0,πœ”0β„Žξ€Έ.(4.7) Using (2.7), (2.11), and (2.16), we can get (β€–β€–1+πœŽβ„Ž)βˆ‡πœ”0β„Žβ€–β€–0β‰€πœ†π‘β€–β€–βˆ‡Μƒπ‘’β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0β€–β€–+(1+πœŽβ„Ž)βˆ‡π‘‡0β€–β€–0,β€–β€–βˆ‡πœ”0β„Žβ€–β€–0β‰€πœ†π‘πœ€4β€–β€–βˆ‡Μƒπ‘’β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0.(4.8) Using the triangle inequality, we have β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0β‰€β€–β€–βˆ‡π‘‡0β€–β€–0+β€–β€–βˆ‡πœ”0β„Žβ€–β€–0β‰€πœ†π‘πœ€4β€–β€–βˆ‡Μƒπ‘’β„Žβ€–β€–0β€–β€–+2βˆ‡π‘‡0β€–β€–0≀5𝐢20π‘πœ†πœ€2+πœ€12(𝜈+πœŽβ„Ž)2β‰€πœ€.(4.9)
Letting π‘£β„Ž=π‘’β„Ž0βˆ—, πœ‘β„Ž=𝑝0β„Ž in (4.5) and using (2.8), we get𝑒(𝜈+πœŽβ„Ž)π‘Žβ„Ž0βˆ—,π‘’β„Ž0βˆ—ξ€Έξ€·=πœ†π‘—π‘‡0β„Ž,π‘’β„Ž0βˆ—ξ€Έ.(4.10) Letting 𝑇0β„Ž=πœ”0β„Ž+𝑇0 and using (2.9), we have β€–β€–(𝜈+πœŽβ„Ž)βˆ‡π‘’β„Ž0βˆ—β€–β€–0≀𝐢20πœ†β€–β€–βˆ‡πœ”0β„Žβ€–β€–0+𝐢0πœ†β€–β€–π‘‡0β€–β€–0≀𝐢20πœ†2π‘β€–β€–βˆ‡Μƒπ‘’β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0+𝐢0πœ†ξ€·1+𝐢0ξ€Έβ€–β€–βˆ‡π‘‡0β€–β€–0≀𝐢20πœ†πœ€.(4.11) Namely, β€–β€–βˆ‡π‘’β„Ž0βˆ—β€–β€–0≀5𝐢20πœ†πœ€.3(𝜈+πœŽβ„Ž)(4.12) Hence, we proved that β„“β„Ž maps ℬ𝑀 to ℬ𝑀. It follows from Brouwer's fixed-point theorem that there exits a solution to system (π’«βˆ—).
To prove the uniqueness, assume that (π‘’β„Ž01,π‘β„Ž01,π‘‡β„Ž01),(π‘’β„Ž02,π‘β„Ž02,π‘‡β„Ž02)βˆˆπ‘‹β„ŽΓ—π‘€β„ŽΓ—π‘Šβ„Ž, and π‘‡β„Ž01|πœ•Ξ©=π‘‡β„Ž02|πœ•Ξ©=𝑇0 are two solutions of (π’«βˆ—). Then, we obtain that𝑒(𝜈+πœŽβ„Ž)π‘Žβ„Ž01βˆ’π‘’β„Ž02,π‘£β„Žξ€Έξ€·π‘βˆ’π‘‘β„Ž01βˆ’π‘β„Ž02,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,π‘’β„Ž01βˆ’π‘’β„Ž02𝑒+π‘β„Ž01βˆ’π‘’β„Ž02,π‘’β„Ž01,π‘£β„Žξ€Έξ€·π‘’+π‘β„Ž02,π‘’β„Ž01βˆ’π‘’β„Ž02,π‘£β„Žξ€Έξ€·π‘—ξ€·π‘‡=πœ†β„Ž01βˆ’π‘‡β„Ž02ξ€Έ,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,(1+πœŽβ„Ž)π‘Žξ€·π‘‡β„Ž01βˆ’π‘‡β„Ž02,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Ž02,π‘‡β„Ž01βˆ’π‘‡β„Ž02,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Ž01βˆ’π‘’β„Ž02,π‘‡β„Ž01,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(4.13) Let π‘£β„Ž=π‘’β„Ž01βˆ’π‘’β„Ž02, πœ‘β„Ž=π‘β„Ž01βˆ’π‘β„Ž02 in the first equation of (4.13), we can get β€–β€–βˆ‡ξ€·π‘’(𝜈+πœŽβ„Ž)β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0β€–β€–βˆ‡ξ€·π‘’β‰€π‘β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0β€–β€–βˆ‡π‘’β„Ž01β€–β€–0+𝐢20πœ†β€–β€–βˆ‡ξ€·π‘‡β„Ž01βˆ’π‘‡β„Ž02ξ€Έβ€–β€–0.(4.14) Setting πœ“β„Ž=π‘‡β„Ž01βˆ’π‘‡β„Ž02 in the second equation of (4.13), we obtain (β€–β€–βˆ‡ξ€·π‘‡1+πœŽβ„Ž)β„Ž01βˆ’π‘‡β„Ž02ξ€Έβ€–β€–0β‰€πœ†π‘β€–β€–βˆ‡ξ€·π‘’β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0β€–β€–βˆ‡π‘‡β„Ž01β€–β€–0.(4.15) By (4.14) and (4.15), we deduce β€–β€–βˆ‡ξ€·π‘’(𝜈+πœŽβ„Ž)β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0β€–β€–β‰€π‘βˆ‡π‘’β„Ž01β€–β€–0β€–β€–βˆ‡(π‘’β„Ž01βˆ’π‘’β„Ž02)β€–β€–0+𝐢20πœ†2π‘β€–β€–βˆ‡π‘‡β„Ž01β€–β€–0β€–β€–βˆ‡(π‘’β„Ž01βˆ’π‘’β„Ž02)β€–β€–0≀5𝐢20π‘πœ†πœ€β€–β€–3(𝜈+πœŽβ„Ž)βˆ‡(π‘’β„Ž01βˆ’π‘’β„Ž02)β€–β€–0+𝐢20πœ†π‘πœ€β€–β€–4πœˆβˆ‡(π‘’β„Ž01βˆ’π‘’β„Ž02)β€–β€–0.(4.16) Using (4.2), we obtain β€–β€–βˆ‡ξ€·π‘’β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0≀25β€–β€–βˆ‡ξ€·π‘’β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0.(4.17) Namely, β€–β€–βˆ‡ξ€·π‘’β„Ž01βˆ’π‘’β„Ž02ξ€Έβ€–β€–0=0.(4.18) By (4.15), we see that β€–βˆ‡(π‘‡β„Ž01βˆ’π‘‡β„Ž02)β€–0=0. Therefore, it follows that (π’«βˆ—) has a unique solution.
Then, we give the prove of (4.1) without using (4.2). Letting π‘£β„Ž=𝑒0β„Ž, πœ‘β„Ž=𝑝0β„Ž in the first equation of (3.1) and using (2.8), we get𝑒(𝜈+πœŽβ„Ž)π‘Ž0β„Ž,𝑒0β„Žξ€Έξ€·=πœ†π‘—π‘‡0β„Ž,𝑒0β„Žξ€Έ.(4.19) Letting 𝑇0β„Ž=πœ”0β„Ž+𝑇0, we have β€–β€–(𝜈+πœŽβ„Ž)βˆ‡π‘’0β„Žβ€–β€–0≀𝐢20πœ†β€–β€–βˆ‡πœ”0β„Žβ€–β€–0+𝐢0πœ†β€–β€–π‘‡0β€–β€–0.(4.20) Letting 𝑇0β„Ž=πœ”0β„Ž+𝑇0, πœ“β„Ž=πœ”0β„Ž in the second equation of (3.1) and using (2.9), we can obtain (1+πœŽβ„Ž)π‘Žξ€·πœ”0β„Ž,πœ”0β„Žξ€Έ=βˆ’πœ†π‘ξ€·π‘’0β„Ž,𝑇0,πœ”0β„Žξ€Έβˆ’(1+πœŽβ„Ž)π‘Žξ€·π‘‡0,πœ”0β„Žξ€Έ.(4.21) Using (2.7), (2.11), and (2.16), we can get (β€–β€–1+πœŽβ„Ž)βˆ‡πœ”0β„Žβ€–β€–0β‰€πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0β€–β€–+(1+πœŽβ„Ž)βˆ‡π‘‡0β€–β€–0,β€–β€–βˆ‡πœ”0β„Žβ€–β€–0β‰€πœ†π‘πœ€4β€–β€–βˆ‡π‘’0β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0.(4.22) By (4.20) and (4.22), we can deduce β€–β€–βˆ‡π‘’0β„Žβ€–β€–0≀(𝜈+πœŽβ„Ž)βˆ’1πœ†πΆ20β€–β€–βˆ‡πœ”0β„Žβ€–β€–0+(𝜈+πœŽβ„Ž)βˆ’1𝐢0πœ†β€–β€–π‘‡0β€–β€–0≀(𝜈+πœŽβ„Ž)βˆ’1ξƒ©πœ†πΆ0‖‖𝑇0β€–β€–0+𝐢20πœ†β€–β€–βˆ‡π‘‡0β€–β€–0+𝐢20πœ†2π‘πœ€4β€–β€–βˆ‡π‘’0β„Žβ€–β€–0ξƒͺ.(4.23) Using (2.16), we get β€–β€–βˆ‡π‘’0β„Žβ€–β€–0≀5𝐢20πœ†πœ€.3(𝜈+πœŽβ„Ž)(4.24) Using (2.7), (2.11), (2.16), and (4.20), we can get β€–β€–βˆ‡πœ”0β„Žβ€–β€–0β‰€πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0β‰€πœ†π‘πœ€4β€–β€–βˆ‡π‘’0β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0≀3πœ€4,β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0β‰€β€–β€–βˆ‡πœ”0β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0β‰€πœ€.(4.25) Therefore, we finish the proof.

Theorem 4.2. Under the assumptions of Lemma 2.3, and 25𝐢20π‘πœ†πœ€3(𝜈+πœŽβ„Ž)2<1,(4.26)(𝑒1β„Ž,𝑇1β„Ž) defined by (3.4) satisfies β€–β€–βˆ‡π‘’1β„Žβ€–β€–0‖‖≀𝛿,βˆ‡π‘‡1β„Žβ€–β€–0≀5πœ€6+πœ†π‘π›Ώπœ€+πœŽβ„Žπœ€,(4.27) where 𝛿≐(103𝐢20πœ†πœ€/48+πœŽβ„Ž(5𝐢20πœ†πœ€/3𝜈))/(7/10)(𝜈+πœŽβ„Ž).

Proof. Letting π‘£β„Ž=𝑒1β„Ž, πœ‘β„Ž=𝑝1β„Ž in the first equation of (3.4) and using (2.8), we get 𝑒(𝜈+πœŽβ„Ž)π‘Ž1β„Ž,𝑒1β„Žξ€Έξ€·π‘’+𝑏1β„Ž,𝑒0β„Ž,𝑒1β„Žξ€Έξ€·π‘’=𝑏0β„Ž,𝑒0β„Ž,𝑒1β„Žξ€Έξ€·π‘’+πœŽβ„Žπ‘Ž0β„Ž,𝑒1β„Žξ€Έξ€·+πœ†π‘—π‘‡0β„Ž,𝑒1β„Žξ€Έ.(4.28) Letting 𝑇0β„Ž=πœ”0β„Ž+𝑇0 and using (2.10), we have β€–β€–(𝜈+πœŽβ„Ž)βˆ‡π‘’1β„Žβ€–β€–0β€–β€–β‰€π‘βˆ‡π‘’1β„Žβ€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–+πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0β€–β€–+π‘βˆ‡π‘’0β„Žβ€–β€–20+𝐢20πœ†β€–β€–βˆ‡πœ”1β„Žβ€–β€–0+𝐢0πœ†β€–β€–π‘‡0β€–β€–0.(4.29) Let 𝑇1β„Ž=πœ”1β„Ž+𝑇0, πœ“β„Ž=πœ”1β„Ž in the second equation of (3.4), we can obtain (1+πœŽβ„Ž)π‘Žξ€·πœ”1β„Ž,πœ”1β„Žξ€Έ=βˆ’πœ†π‘ξ€·π‘’0β„Ž,𝑇0,πœ”1β„Žξ€Έβˆ’πœ†π‘ξ€·π‘’1β„Ž,𝑇0β„Ž,πœ”1β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,𝑇0β„Ž,πœ”1β„Žξ€Έ+πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,πœ”1β„Žξ€Έβˆ’(1+πœŽβ„Ž)π‘Žξ€·π‘‡0,πœ”1β„Žξ€Έ.(4.30) Using (2.11) and (2.16), we can get (β€–β€–1+πœŽβ„Ž)βˆ‡πœ”1β„Žβ€–β€–0β‰€πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0+πœ†π‘β€–β€–βˆ‡π‘’1β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0β€–β€–+πœŽβ„Žβˆ‡π‘‡0β„Žβ€–β€–0β€–β€–+(1+πœŽβ„Ž)βˆ‡π‘‡0β€–β€–0,β€–β€–(4.31)βˆ‡πœ”1β„Žβ€–β€–0β‰€πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0+πœ†π‘β€–β€–βˆ‡π‘’1β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0β€–β€–+πœŽβ„Žβˆ‡π‘‡0β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0.(4.32) Using (4.29), we get β€–β€–(𝜈+πœŽβ„Ž)βˆ‡π‘’1β„Žβ€–β€–0β€–β€–β‰€π‘βˆ‡π‘’1β„Žβ€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–+πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0β€–β€–+π‘βˆ‡π‘’0β„Žβ€–β€–20+𝐢20πœ†ξ‚€πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0+πœ†π‘β€–β€–βˆ‡π‘’1β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+πœ†π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0ξ€Έ+𝐢0πœ†β€–β€–π‘‡0β€–β€–0.ξ‚€β€–β€–πœˆ+πœŽβ„Žβˆ’π‘βˆ‡π‘’0β„Žβ€–β€–0βˆ’πΆ20πœ†2π‘β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0ξ‚β€–β€–βˆ‡π‘’1β„Žβ€–β€–0β€–β€–β‰€π‘βˆ‡π‘’0β„Žβ€–β€–20β€–β€–+πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0+𝐢20πœ†2π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0+𝐢20πœ†2π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+𝐢20πœ†β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0+𝐢20πœ†β€–β€–βˆ‡π‘‡0β€–β€–0+𝐢0πœ†β€–β€–π‘‡0β€–β€–0.(4.33) Using (2.16), (4.26), and Theorem 4.2, we can obtain 7β€–β€–10(𝜈+πœŽβ„Ž)βˆ‡π‘’1β„Žβ€–β€–0≀25𝑁𝐢40πœ†2πœ€29(𝜈+πœŽβ„Ž)2+πœŽβ„Ž5𝐢20πœ†πœ€3+(𝜈+πœŽβ„Ž)10𝑁𝐢40πœ†3πœ€23+(𝜈+πœŽβ„Ž)3𝐢20πœ†2πœ€22≀103𝐢20πœ†πœ€48+πœŽβ„Ž5𝐢20πœ†πœ€.3(𝜈+πœŽβ„Ž)(4.34) Namely, β€–β€–βˆ‡π‘’1β„Žβ€–β€–0≀103𝐢20ξ€·πœ†πœ€/48+πœŽβ„Ž5𝐢20ξ€Έπœ†πœ€/3𝜈(7/10)(𝜈+πœŽβ„Ž)≐𝛿.(4.35)
Using (2.16), (4.31), and (4.35), we can getβ€–β€–βˆ‡πœ”1β„Žβ€–β€–0≀10𝑁𝐢20πœ†2πœ€23(𝜈+πœŽβ„Ž)+πœ†πœ€π‘π›Ώπœ€+πœŽβ„Žπœ€+4β‰€πœ€3+πœ€4+πœ†=π‘π›Ώπœ€+πœŽβ„Žπœ€7πœ€12+πœ†π‘π›Ώπœ€+πœŽβ„Žπœ€.(4.36) Using the triangle inequality, we can get β€–β€–βˆ‡π‘‡1β„Žβ€–β€–0β‰€β€–β€–βˆ‡πœ”1β„Žβ€–β€–0+β€–β€–βˆ‡π‘‡0β€–β€–0≀5πœ€6+πœ†π‘π›Ώπœ€+πœŽβ„Žπœ€.(4.37) Therefore, we finish the proof.

5. Error Analysis

In this section, we establish the 𝐻1 and 𝐿2-bounds of the error π‘’βˆ’π‘’π‘–β„Ž, π‘‡βˆ’π‘‡π‘–β„Ž, 𝑖=0,1 and 𝐿2-bound of the error π‘βˆ’π‘π‘–β„Ž, 𝑖=0,1. In order to obtain the error estimates, we define the Galerkin projection (π‘…β„Ž,π‘„β„Ž)=(π‘…β„Ž(𝑒,𝑝),π‘„β„Ž(𝑒,𝑝))∢(𝑋,𝑀)β†’(π‘‹β„Ž,π‘€β„Ž), such thatπ‘Žξ€·π‘…β„Žβˆ’π‘’,π‘£β„Žξ€Έξ€·π‘„βˆ’π‘‘β„Žβˆ’π‘,π‘£β„Žξ€Έξ€·π‘ž+π‘‘β„Ž,π‘…β„Žξ€Έξ€·π‘£βˆ’π‘’=0,βˆ€(𝑒,𝑝)∈(𝑋,𝑀),β„Ž,π‘žβ„Žξ€Έβˆˆξ€·π‘‹β„Ž,π‘€β„Žξ€Έ.(5.1)

Lemma 5.1 (see [37, 38]). The Galerkin projection (π‘…β„Ž,π‘„β„Ž) satisfies β€–β€–π‘…β„Žβ€–β€–βˆ’π‘’0ξ€·β€–β€–βˆ‡ξ€·π‘…+β„Žβ„Žξ€Έβ€–β€–βˆ’π‘’0+β€–β€–π‘„β„Žβ€–β€–βˆ’π‘0ξ€Έβ‰€πΆβ„Žπ‘Ÿ+1ξ€·πœˆβ€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿξ€Έ,π‘Ÿ=1,2.(5.2)

Lemma 5.2 (see [7]). There exits Μƒπ‘Ÿβ„ŽβˆΆπ‘Šβ†’π‘Šβ„Ž for all πœ“βˆˆπ‘Š holds that ξ€·βˆ‡ξ€·πœ“βˆ’Μƒπ‘Ÿβ„Žπœ“ξ€Έ,βˆ‡πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Šβ„Žξ€œ,(5.3)Ξ©ξ€·πœ“βˆ’Μƒπ‘Ÿβ„Žπœ“ξ€Έβ€–β€–π‘‘π‘₯=0,βˆ‡Μƒπ‘Ÿβ„Žπœ“β€–β€–0β‰€β€–βˆ‡πœ“β€–0.(5.4) When πœ“βˆˆπ‘Šπ‘˜,π‘ž(Ξ©)(1β‰€π‘žβ‰€βˆž), there holds β€–β€–πœ“βˆ’Μƒπ‘Ÿβ„Žπœ“β€–β€–βˆ’π‘ ,π‘žβ‰€πΆβ„Žπ‘˜+𝑠||πœ“||π‘˜,π‘ž,βˆ’1β‰€π‘ β‰€π‘š,0β‰€π‘˜β‰€π‘Ÿ+1.(5.5) There exits π‘Ÿβ„ŽβˆΆπ‘Š0β†’π‘Š0β„Ž for all πœ“βˆˆπ‘Š0 holds that ξ€·βˆ‡ξ€·πœ“βˆ’π‘Ÿβ„Žπœ“ξ€Έ,βˆ‡πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž,β€–β€–βˆ‡π‘Ÿβ„Žπœ“β€–β€–0β‰€β€–βˆ‡πœ“β€–0.(5.6) When πœ“βˆˆπ‘Šπ‘Ÿ,π‘ž(Ξ©)(1β‰€π‘žβ‰€βˆž), there holds β€–β€–πœ“βˆ’π‘Ÿβ„Žπœ“β€–β€–βˆ’π‘ ,π‘žβ‰€πΆβ„Žπ‘˜+𝑠||πœ“||π‘Ÿ,π‘ž,βˆ’1β‰€π‘ β‰€π‘Ÿ,0β‰€π‘˜β‰€π‘Ÿ+1.(5.7)

Lemma 5.3 (see [7]). If (A1)~(A3) hold and (𝑒,𝑝,𝑇)βˆˆπ»π‘Ÿ+1(Ξ©)Γ—π»π‘Ÿ(Ξ©)Γ—π»π‘Ÿ+1(Ξ©) and (π‘’β„Ž,π‘ƒβ„Ž,π‘‡β„Ž) are the solution of problem (π’«ξ…ž) and (π’«ξ…žξ…ž), respectively, then there holds that β€–β€–βˆ‡(π‘’βˆ’π‘’β„Ž)β€–β€–0+β€–β€–π‘βˆ’π‘β„Žβ€–β€–0+β€–β€–βˆ‡(π‘‡βˆ’π‘‡β„Ž)β€–β€–0β‰€πΆβ„Žπ‘Ÿξ€·β€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿ+β€–π‘‡β€–π‘Ÿ+1ξ€Έ.(5.8)

Lemma 5.4. Under the assumptions of Lemma 2.3, (π‘’β„Ž,π‘β„Ž,π‘‡β„Ž) is the solution of (3.1), (𝑒0β„Ž,𝑝0β„Ž,𝑇0β„Ž) defined by (3.4), then there hold β€–β€–βˆ‡(π‘’β„Žβˆ’π‘’0β„Ž)β€–β€–0≀50𝜎𝐢20πœ†β„Ž+21𝜈10𝐢20πœ†πœŽπœ€β„Ž,β€–β€–7πœˆβˆ‡(π‘‡β„Žβˆ’π‘‡0β„Ž)β€–β€–0≀2πœŽβ„Žπœ€+πœŽβ„Žπœ€,𝛽‖‖𝑝14πœˆβ„Žβˆ’π‘0β„Žβ€–β€–0≀95πœŽβ„ŽπΆ20πœ†πœ€+21𝜈19πœŽβ„ŽπΆ20πœ†πœ€7+πœŽβ„ŽπΆ20πœ†+πœŽβ„ŽπΆ20πœ†.14𝜈(5.9)

Proof. Subtracting (3.1) from (2.5) we get the error equations, namely (π‘’β„Žβˆ’π‘’0β„Ž,π‘β„Žβˆ’π‘0β„Ž,π‘‡β„Žβˆ’π‘‡0β„Ž) satisfy ξ€·π‘’πœˆπ‘Žβ„Žβˆ’π‘’0β„Ž,π‘£β„Žξ€Έξ€·π‘’βˆ’πœŽβ„Žπ‘Ž0β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,π‘’β„Žβˆ’π‘’0β„Žξ€Έξ€·π‘βˆ’π‘‘β„Žβˆ’π‘0β„Ž,π‘£β„Žξ€Έξ€·π‘’+𝑏0β„Ž,π‘’β„Žβˆ’π‘’0β„Ž,π‘£β„Žξ€Έξ€·π‘’+π‘β„Žβˆ’π‘’0β„Ž,π‘’β„Ž,π‘£β„Žξ€Έξ€·π‘—ξ€·π‘‡=πœ†β„Žβˆ’π‘‡0β„Žξ€Έ,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,π‘Žξ€·π‘‡β„Žβˆ’π‘‡0β„Ž,πœ“β„Žξ€Έβˆ’πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Žβˆ’π‘’0β„Ž,π‘‡β„Ž,πœ“β„Žξ€Έ+𝑏𝑒0β„Ž,π‘‡β„Žβˆ’π‘‡0β„Ž,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(5.10) Letting π‘£β„Ž=π‘’β„Žβˆ’π‘’0β„Ž,πœ‘β„Ž=π‘β„Žβˆ’π‘ƒ0β„Ž in the first equation of (5.10) and using (2.11), (2.8), and (A1), we can get πœˆβ€–β€–βˆ‡ξ€·π‘’β„Žβˆ’π‘’0β„Žξ€Έβ€–β€–0β€–β€–β‰€πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡ξ€·π‘’+π‘β„Žβˆ’π‘’0β„Žξ€Έβ€–β€–0β€–β€–βˆ‡π‘’β„Žβ€–β€–0+𝐢20πœ†β€–β€–βˆ‡ξ€·π‘‡β„Žβˆ’π‘‡0β„Žξ€Έβ€–β€–0.(5.11) Hence, we deduce ξ€·β€–β€–πœˆβˆ’π‘βˆ‡π‘’β„Žβ€–β€–0ξ€Έβ€–β€–βˆ‡ξ€·π‘’β„Žβˆ’π‘’0β„Žξ€Έβ€–β€–0β€–β€–β‰€πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0+𝐢20πœ†β€–β€–βˆ‡ξ€·π‘‡β„Žβˆ’π‘‡0β„Žξ€Έβ€–β€–0.(5.12) Letting πœ“β„Ž=π‘‡β„Žβˆ’π‘‡0β„Ž in the second equation of (5.10) and using (2.9), we obtain π‘Žξ€·π‘‡β„Žβˆ’π‘‡0β„Ž,π‘‡β„Žβˆ’π‘‡0β„Žξ€Έ+πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,π‘‡β„Žβˆ’π‘‡0β„Žξ€Έ+πœ†π‘ξ€·π‘’β„Žβˆ’π‘’0β„Ž,π‘‡β„Ž,π‘‡β„Žβˆ’π‘‡0β„Žξ€Έ=0.(5.13) Using (2.11), we can get β€–β€–βˆ‡(π‘‡β„Žβˆ’π‘‡0β„Ž)β€–β€–0β€–β€–β‰€πœŽβ„Žβˆ‡π‘‡0β„Žβ€–β€–0+πœ†π‘β€–β€–βˆ‡(π‘’β„Žβˆ’π‘’0β„Ž)β€–β€–0β€–β€–βˆ‡π‘‡β„Žβ€–β€–0.(5.14) By (5.12), we deduce ξ€·β€–β€–πœˆβˆ’π‘βˆ‡π‘’β„Žβ€–β€–0ξ€Έβ€–β€–βˆ‡ξ€·π‘’β„Žβˆ’π‘’0β„Žξ€Έβ€–β€–0β€–β€–β‰€πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0+𝐢20β€–β€–πœ†πœŽβ„Žβˆ‡π‘‡0β„Žβ€–β€–0+𝐢20πœ†2π‘β€–β€–βˆ‡(π‘’β„Žβˆ’π‘’0β„Ž)β€–β€–0β€–β€–βˆ‡π‘‡β„Žβ€–β€–0.(5.15) Using (4.1), we can obtain ξ‚€β€–β€–πœˆβˆ’π‘βˆ‡π‘’β„Žβ€–β€–0βˆ’3πœˆξ‚β€–β€–βˆ‡ξ€·π‘’32β„Žβˆ’π‘’0β„Žξ€Έβ€–β€–0β€–β€–β‰€πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0+𝐢20β€–β€–πœ†πœŽβ„Žβˆ‡π‘‡0β„Žβ€–β€–0≀5πœŽβ„ŽπΆ20πœ†πœ€3(𝜈+πœŽβ„Ž)+𝐢20πœ†πœŽβ„Žπœ€.(5.16) By using (2.16) and (2.17), there holds β€–β€–πœˆβˆ’π‘βˆ‡π‘’β„Žβ€–β€–0βˆ’3𝜈β‰₯327𝜈.10(5.17) Therefore, we can deduce β€–β€–βˆ‡(π‘’β„Žβˆ’π‘’0β„Ž)β€–β€–0≀50πœŽβ„ŽπΆ20πœ†πœ€+21𝜈(𝜈+πœŽβ„Ž)10𝐢20πœ†πœŽβ„Žπœ€.7𝜈(5.18) By (5.14) and (5.18), we can have β€–β€–βˆ‡(π‘‡β„Žβˆ’π‘‡0β„Ž)β€–β€–0β‰€πœŽβ„Žπœ€+πœ†ξƒ©π‘πœ€50πœŽβ„ŽπΆ20πœ†πœ€+21𝜈(𝜈+πœŽβ„Ž)10𝐢20πœ†πœŽβ„Žπœ€ξƒͺ7πœˆβ‰€2πœŽβ„Žπœ€+πœŽβ„Žπœ€.14𝜈(5.19) Letting πœ‘β„Ž=0, π‘£β„Ž=π‘’β„Žβˆ’π‘’0β„Ž in the first equation of (5.10) and using (2.3), we have π›½β€–β€–π‘β„Žβˆ’π‘0β„Žβ€–β€–0β€–β€–β‰€πœˆβˆ‡(π‘’β„Žβˆ’π‘’0β„Ž)β€–β€–0β€–β€–+πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0β€–β€–+π‘βˆ‡(π‘’β„Žβˆ’π‘’0β„Ž)β€–β€–0β€–β€–βˆ‡π‘’β„Žβ€–β€–0+𝐢20πœ†β€–β€–βˆ‡(π‘‡β„Žβˆ’π‘‡0β„Ž)β€–β€–0≀50πœŽβ„ŽπΆ20πœ†πœ€+21(𝜈+πœŽβ„Ž)10πœŽβ„ŽπΆ20πœ†πœ€7+5πœŽβ„ŽπΆ20πœ†πœ€+3𝜈10πœŽβ„ŽπΆ20πœ†πœ€+21𝜈2πœŽβ„ŽπΆ20πœ†πœ€7+πœŽβ„ŽπΆ20πœ†πœ€+πœŽβ„ŽπΆ20πœ†+πœŽβ„ŽπΆ20πœ†β‰€14𝜈95πœŽβ„ŽπΆ20πœ†πœ€+21(𝜈+πœŽβ„Ž)19πœŽβ„ŽπΆ20πœ†πœ€7+πœŽβ„ŽπΆ20πœ†+πœŽβ„ŽπΆ20πœ†.14𝜈(5.20) Hence, we finish the proof.

Theorem 5.5. Under the assumptions of Lemmas 2.3 and 5.3, the following inequality β€–β€–βˆ‡(π‘’βˆ’π‘’0β„Ž)β€–β€–0+β€–β€–π‘βˆ’π‘0β„Žβ€–β€–0+β€–β€–βˆ‡(π‘‡βˆ’π‘‡0β„Ž)β€–β€–0β‰€πΆβ„Žπ‘Ÿξ€·β€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿ+β€–π‘‡β€–π‘Ÿ+1ξ€Έ+πΆβ„Ž,(5.21) holds, where 𝐢 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 π‘’βˆˆπ»2(Ξ©)βˆ©π‘‹, πœ”βˆˆπ‘Š0, πœ“βˆˆπ»2(Ξ©)βˆ©π‘Š0, there hold that |||π‘ξ€·π‘’βˆ’π‘…β„Žξ€Έ|||β€–β€–,πœ”,πœ“β‰€πΆπ‘’βˆ’π‘…β„Žβ€–β€–0β€–π’œπœ”β€–0β€–βˆ‡πœ“β€–0|||,(5.22)𝑏𝑒,π‘‡βˆ’Μƒπ‘Ÿβ„Žξ€Έ|||𝑇,πœ“β‰€πΆβ€–π’œπ‘’β€–0β€–β€–π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡β€–β€–0β€–βˆ‡πœ“β€–0.(5.23)

Proof. Letting πœ”=(πœ”,0)𝑇, we have π‘ξ€·π‘’βˆ’π‘…β„Žξ€Έξ€·,πœ”,πœ“=π‘π‘’βˆ’π‘…β„Ž,ξ€Έ.πœ”,πœ“(5.24) Using (2.25), we can deduce (5.22). Because π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡βˆˆπ‘Š0, (5.23) holds.

Theorem 5.7. Under the assumptions of Lemmas 2.3 and 5.3, the following inequality: β€–β€–π‘’βˆ’π‘’0β„Žβ€–β€–0+β€–β€–π‘‡βˆ’π‘‡0β„Žβ€–β€–0β‰€πΆβ„Žπ‘Ÿ+1ξ€·β€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿ+β€–π‘‡β€–π‘Ÿ+1ξ€Έ+πΆβ„Ž,(5.25) holds, where 𝐢 is a positive constant.

Proof. Subtracting (3.1) from (2.4) we get the error equations, namely, ξ€·πœˆπ‘Žπ‘’βˆ’π‘’0β„Ž,π‘£β„Žξ€Έξ€·π‘’βˆ’πœŽβ„Žπ‘Ž0β„Ž,π‘£β„Žξ€Έξ€·+π‘π‘’βˆ’π‘’0β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έξ€·+𝑏𝑒,π‘’βˆ’π‘’0β„Ž,π‘£β„Žξ€Έξ€·βˆ’π‘‘π‘βˆ’π‘0β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,π‘’βˆ’π‘’0β„Žξ€Έξ€·π‘—ξ€·=πœ†π‘‡βˆ’π‘‡0β„Žξ€Έ,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,π‘Žξ€·π‘‡βˆ’π‘‡0β„Ž,πœ“β„Žξ€Έβˆ’πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’βˆ’π‘’0β„Ž,𝑇,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,π‘‡βˆ’π‘‡0β„Ž,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(5.26) Letting 𝑒0β„Ž=π‘…β„Žβˆ’π‘’0β„Ž, πœ‚0β„Ž=π‘„β„Žβˆ’π‘0β„Ž, πœ‰0β„Ž=Μƒπ‘Ÿβ„Žπ‘‡βˆ’π‘‡0β„Ž and using (5.1) and (5.3), we can get ξ€·π‘’πœˆπ‘Ž0β„Ž,π‘£β„Žξ€Έξ€·π‘’βˆ’πœŽβ„Žπ‘Ž0β„Ž,π‘£β„Žξ€Έξ€·+π‘π‘’βˆ’π‘’0β„Ž,𝑒0β„Ž,π‘£β„Žξ€Έξ€·+𝑏𝑒,π‘’βˆ’π‘’0β„Ž,π‘£β„Žξ€Έξ€·πœ‚βˆ’π‘‘0β„Ž,π‘£β„Žξ€Έξ€·πœ‘+π‘‘β„Ž,𝑒0β„Žξ€Έξ€·π‘—ξ€·=πœ†π‘‡βˆ’π‘‡0β„Žξ€Έ,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž,π‘Žξ€·πœ‰β„Ž,πœ“β„Žξ€Έβˆ’πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’βˆ’π‘’0β„Ž,𝑇,πœ“β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,π‘‡βˆ’π‘‡0β„Ž,πœ“β„Žξ€Έ=0,βˆ€πœ“β„Žβˆˆπ‘Š0β„Ž.(5.27) Taking π‘£β„Ž=𝑒0β„Ž, πœ‘β„Ž=πœ‚0β„Ž in the first equation of (5.27), we obtain ξ€·π‘’πœˆπ‘Ž0β„Ž,𝑒0β„Žξ€Έξ€·π‘’βˆ’πœŽβ„Žπ‘Ž0β„Ž,𝑒0β„Žξ€Έξ€·π‘’+𝑏0β„Ž,𝑒0β„Ž,𝑒0β„Žξ€Έξ€·+π‘π‘’βˆ’π‘…β„Ž,𝑒0β„Ž,𝑒0β„Žξ€Έξ€·+𝑏𝑒,π‘’βˆ’π‘…β„Ž,𝑒0β„Žξ€Έξ€·π‘—ξ€·=πœ†π‘‡βˆ’π‘‡0β„Žξ€Έ,𝑒0β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‹β„Ž,πœ‘β„Žβˆˆπ‘€β„Ž.(5.28) Using (2.10) and (A1), we deduce ξ€·β€–β€–πœˆβˆ’π‘βˆ‡π‘’0β„Žβ€–β€–0ξ€Έβ€–β€–βˆ‡π‘’0β„Žβ€–β€–20≀||π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑒,𝑒0β„Žξ€Έ||+||𝑏𝑒0β„Ž,π‘’βˆ’π‘…β„Ž,𝑒0β„Žξ€Έ||+||πœ†ξ€·π‘—ξ€·π‘‡βˆ’π‘‡0β„Žξ€Έ,𝑒0β„Žξ€Έ||+||ξ€·π‘’πœŽβ„Žπ‘Ž0β„Ž,𝑒0β„Žξ€Έ||ξ€·β‰€π‘β€–βˆ‡π‘’β€–0+β€–β€–βˆ‡π‘’0β„Žβ€–β€–0ξ€Έβ€–β€–βˆ‡(π‘’βˆ’π‘…β„Ž)β€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0+𝐢20πœ†β€–β€–βˆ‡(π‘‡βˆ’π‘‡0β„Ž)β€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–+πœŽβ„Žβˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0.(5.29) Using Theorem 2.1, (2.16), (4.1), and (5.2), we can obtain β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β‰€πΆβ„Žπ‘Ÿ+1ξ€·β€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿ+β€–π‘‡β€–π‘Ÿ+1ξ€Έ+πΆβ„Ž.(5.30) Taking πœ“β„Ž=πœ‰0β„Ž in the second equation of (5.27) and using (2.9) we have π‘Žξ€·πœ‰0β„Ž,πœ‰0β„Žξ€Έβˆ’πœŽβ„Žπ‘Žξ€·π‘‡0β„Ž,πœ‰0β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έ+πœ†π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑇,πœ‰0β„Žξ€Έ+πœ†π‘ξ€·π‘’0β„Ž,𝑇,πœ‰0β„Žξ€Έ=0.(5.31) By (2.9), we have 𝑏𝑒0β„Ž,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έ+π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑇,πœ‰0β„Žξ€Έ+𝑏𝑒0β„Ž,𝑇,πœ‰0β„Žξ€Έ=βˆ’π‘ξ€·π‘’0β„Ž,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έβˆ’π‘ξ€·π‘’βˆ’π‘…β„Ž,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έ+𝑏𝑒,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έ+π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑇,πœ‰0β„Žξ€Έ+𝑏𝑒0β„Ž,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έ+𝑏𝑒0β„Ž,πœ‰0β„Ž,πœ‰0β„Žξ€Έ+𝑏𝑒0β„Ž,𝑇0β„Ž,πœ‰0β„Žξ€Έ.(5.32) Letting 𝑇=πœ”+𝑇0, πœ”βˆˆπ‘Š0 and using Lemma 5.6, we can get |||𝑏𝑒0β„Ž,π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡,πœ‰0β„Žξ€Έ+π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑇,πœ‰0β„Žξ€Έ+π‘ξ€·π‘’β„Ž,𝑇,πœ‰0β„Žξ€Έ|||β‰€π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡(π‘‡βˆ’Μƒπ‘Ÿβ„Žβ€–β€–π‘‡)0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0+π‘β€–β€–βˆ‡(π‘’βˆ’π‘…β„Ž)β€–β€–0β€–β€–βˆ‡(π‘‡βˆ’Μƒπ‘Ÿβ„Žβ€–β€–π‘‡)0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0+π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡(π‘‡βˆ’Μƒπ‘Ÿβ„Žβ€–β€–π‘‡)0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0+π‘β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β€–β€–βˆ‡π‘‡0β„Žβ€–β€–0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0+πΆβ€–π’œπ‘’β€–0β€–β€–π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡β€–β€–0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0β€–β€–+πΆπ‘’βˆ’π‘…β„Žβ€–β€–0β€–π’œπœ”β€–0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0β€–β€–+πΆβˆ‡(π‘’βˆ’π‘…β„Ž)β€–β€–0β€–β€–βˆ‡π‘‡0β€–β€–0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0.(5.33) By assumption (A2), letting πœ€<β„Ž and using Lemma 5.1 and (2.16), (4.26), and (5.33), we can deduce β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0β‰€πΆβ„Žπ‘Ÿ+1ξ€·β€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿ+β€–π‘‡β€–π‘Ÿ+1ξ€Έ+πΆβ„Ž.(5.34) Hence, we have β€–β€–π‘‡βˆ’π‘‡0β„Žβ€–β€–0β‰€β€–β€–π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡β€–β€–0+β€–β€–πœ‰0β„Žβ€–β€–0β‰€β€–β€–π‘‡βˆ’Μƒπ‘Ÿβ„Žπ‘‡β€–β€–0+𝐢0β€–β€–βˆ‡πœ‰0β„Žβ€–β€–0β‰€πΆβ„Žπ‘Ÿ+1ξ€·β€–π‘’β€–π‘Ÿ+1+β€–π‘β€–π‘Ÿ+β€–π‘‡β€–π‘Ÿ+1ξ€Έ+πΆβ„Ž.(5.35)
By (2.10) and (2.25), we can deduce||π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑒,𝑒0β„Žξ€Έ||+||𝑏𝑒0β„Ž,π‘’βˆ’π‘…β„Ž,𝑒0β„Žξ€Έ||≀||π‘ξ€·π‘’βˆ’π‘…β„Ž,𝑒,𝑒0β„Žξ€Έ||+||𝑏𝑒,π‘’βˆ’π‘…β„Ž,𝑒0β„Žξ€Έ||+||π‘ξ€·π‘’βˆ’π‘…β„Ž,π‘’βˆ’π‘…β„Ž,𝑒0β„Žξ€Έ||+||π‘ξ€·π‘’β„Ž,π‘’βˆ’π‘…β„Ž,𝑒0β„Žξ€Έ||β‰€πΆβ€–π’œπ‘’β€–0β€–β€–π‘’βˆ’π‘…β„Žβ€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0ξ€·β€–β€–βˆ‡ξ€·+π‘π‘’βˆ’π‘…β„Žξ€Έβ€–β€–0+β€–β€–βˆ‡π‘’0β„Žβ€–β€–0ξ€Έβ€–β€–βˆ‡ξ€·π‘’βˆ’π‘…β„Žξ€Έβ€–β€–0β€–β€–βˆ‡π‘’0β„Žβ€–β€–0β‰€πΆβ„Ž2β€–β€–βˆ‡π‘’0β„Žβ€–β€–0.(5.36) Using (5.29), we can obtain ξ€·β€–β€–πœˆβˆ’