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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 570980, 18 pages
http://dx.doi.org/10.1155/2011/570980
Research Article

An Optimal Error Estimates of 𝐻1-Galerkin Expanded Mixed Finite Element Methods for Nonlinear Viscoelasticity-Type Equation

1School of Management Science, Qufu Normal University, Rizhao, Shandong 276800, China
2School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261000, China
3School of Mathematical Sciences, Shandong Normal University, Jinan, Shandong 250014, China

Received 30 March 2011; Revised 14 August 2011; Accepted 21 August 2011

Academic Editor: Ben T.Β Nohara

Copyright Β© 2011 Haitao Che et al. 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

We investigate a 𝐻1-Galerkin mixed finite element method for nonlinear viscoelasticity equations based on 𝐻1-Galerkin method and expanded mixed element method. The existence and uniqueness of solutions to the numerical scheme are proved. A priori error estimation is derived for the unknown function, the gradient function, and the flux.

1. Introduction

Consider the following nonlinear viscoelasticity-type equation: π‘’π‘‘π‘‘ξ€·π‘Žβˆ’βˆ‡β‹…(π‘₯,𝑒)βˆ‡π‘’π‘‘ξ€Έ+𝑏(π‘₯,𝑒)βˆ‡π‘’=𝑓(π‘₯,𝑑),(π‘₯,𝑑)βˆˆΞ©Γ—π½,𝑒(π‘₯,𝑑)=0,(π‘₯,𝑑)βˆˆπœ•Ξ©Γ—π½,𝑒(π‘₯,0)=𝑒0(𝑒π‘₯),π‘₯∈Ω,𝑑(π‘₯,0)=𝑒1(π‘₯),π‘₯∈Ω,(1.1) where Ξ© is a convex polygonal domain in 𝑅2 with the Lipschitz continuous boundary πœ•Ξ©, 𝐽=(0,𝑇] is the time interval with 0<𝑇<∞, and 𝑒0(π‘₯) and 𝑒1(π‘₯) are, respectively, the initial data functions defined on Ξ©. The deformation of viscoelastic solid under the external loads is usually considered by means of this viscoelastic model [1–4], and the problem has a unique sufficiently smooth solution with the regularity condition provided that the given data 𝑒0(π‘₯), 𝑒1(π‘₯), π‘Ž(𝑒), 𝑏(𝑒), and 𝑓 are sufficiently smooth [5].

For problem (1.1), by adopting finite element method, Lin et al. [6] established the convergence of the finite element approximations to solutions of Sobolev and viscoelasticity type of equations via Ritz-Volterra projection and an optimal-order error estimates in 𝐿𝑝 (2≀𝑝<∞). Latter, Lin and Zhang [7] presented a direct analysis for global superconvergence for this problem without using the Ritz projection or its modified forms. Jin et al. [8] and Shi et al. [9] employed the Wilson nonconforming finite element and a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes to solve viscoelasticity-type equations, and the global superconvergence estimations were obtained by means of post-processing technique. Since the estimation of flux βˆ‡π‘’ by the unknown scalar 𝑒 is usually indirect, thus the quantity of calculation of the finite element method is relatively large.

As an efficient strategy, mixed finite element methods received much attention in solving partial differential equation in recent decades [10–16]. Compared with finite element methods, mixed finite element methods can obtain the unknown scalar 𝑒 and its flux βˆ‡π‘’ directly, and; hence, it can decrease smoothness of solution space. However, the LBB assumption is needed in the approximating subspaces and; hence, confines the choice of finite element spaces.

On the base of the mixed finite element methods, Pani [17] proposed a new mixed finite element method, called the 𝐻1-Galerkin mixed finite element procedure, to solve a mixed system in unknown scalar and its flux. Compared with the standard mixed finite methods, the new mixed finite element method does not require the LBB condition, and a better order of convergence for the flux in 𝐿2 norm can be obtained if an extra regularity on the solution holds. Recently, 𝐻1-Galerkin mixed finite element methods were applied to differential equations [18–22]. However, the assumption needed for this method is not suitable for the nonlinear equations and equations with a small tensor. To overcome this, Chen and Wang [23] proposed 𝐻1-Galerkin expanded mixed finite element methods which combines the 𝐻1-Galerkin formulation and the expanded mixed finite element methods [24] to deal with a nonlinear parabolic equation in porous medium flow. This method can compute the scalar unknown, its gradient, and its flux directly. Hence, it is suitable to the case where the coefficient of the differential equation is a small tensor and cannot be inverted. Motivated by this, we establish an 𝐻1-Galerkin expanded mixed finite element method for the viscoelasticity-type equations.

The remainder of this paper is organized as follows. In Section 2, we first establish the equivalence between viscoelasticity-type equations and their weak formulation by using the 𝐻1-Galerkin expanded mixed finite element methods and then discuss the existence and uniqueness of the formulation. In Section 3, we show that the 𝐻1-Galerkin expanded mixed finite element method has the same convergence rate as that of the classical mixed finite element methods without requiring the LBB consistency condition.

Throughout this paper, we use 𝐻 to denote the space 𝐻(div,Ξ©)={𝐯∈(𝐿2(Ξ©))π‘‘βˆΆβˆ‡β‹…π―βˆˆπΏ2(Ξ©)} with norm ‖𝐯‖𝐇(div;Ξ©)=(‖𝐯‖2+β€–βˆ‡β‹…π―β€–2)1/2 and 𝐻10(Ξ©)={π‘€βˆˆπ»1(Ξ©)βˆΆπ‘€=0onπœ•Ξ©}. For theoretical analysis, we also need the following assumptions on the functions involved in problem (1.1).

Assumption 1.1. (1) There exist constants a1 and a2 such that 0<π‘Ž1β‰€π‘Ž(π‘₯,𝑒),𝑏(π‘₯,𝑒)β‰€π‘Ž2.
(2) The functions π‘Ž(π‘₯,𝑒),𝑏(π‘₯,𝑒),π‘Žπ‘’(π‘₯,𝑒), and 𝑏𝑒(π‘₯,𝑒) are Lipschitz continuous with respect to 𝑒, and there exists 𝐢1>0 such that |πœ•π‘Ž/πœ•π‘’|+|πœ•π‘/πœ•π‘’|+|πœ•2π‘Ž/πœ•π‘’2|+|πœ•2𝑏/πœ•π‘’2|≀𝐢1.

2. 𝐻1-Galerkin Expanded Mixed Finite Element Discrete Scheme

2.1. Weak Formulation

To define the 𝐻1-Galerkin expanded mixed finite element procedure, we introduce vector 𝐩=π‘Ž(π‘₯,𝑒)βˆ‡π‘’π‘‘+𝑏(π‘₯,𝑒)βˆ‡π‘’,𝝈=βˆ‡π‘’,(2.1) and split (1.1) into a first-order system as follows: π‘’π‘‘π‘‘βˆ’βˆ‡β‹…π©=𝑓,𝝈=βˆ‡π‘’,𝐩=π‘Ž(𝑒)πˆπ‘‘+𝑏(𝑒)𝝈,𝝈(π‘₯,0)=βˆ‡π‘’0𝝈(π‘₯),𝑑(π‘₯,0)=βˆ‡π‘’1𝑒(π‘₯),𝐩(π‘₯,0)=π‘Ž0ξ€Έβˆ‡π‘’1𝑒(π‘₯)+𝑏0ξ€Έβˆ‡π‘’0(π‘₯).(2.2) Then by Green's formula we can further define the following weak formulation of problem (2.2): find (𝑒,𝝈,𝐩)∈𝐻10(Ξ©)×𝐻(div,Ξ©)×𝐻(div,Ξ©) such that ξ€·πˆπ‘‘π‘‘ξ€Έ+,πͺ(βˆ‡β‹…π©,βˆ‡β‹…πͺ)=βˆ’(𝑓,βˆ‡β‹…πͺ),βˆ€πͺ∈𝐻(div,Ξ©),(𝝈,βˆ‡π‘£)=(βˆ‡π‘’,βˆ‡π‘£),βˆ€π‘£βˆˆπ»10ξ€·(Ξ©),(𝐩,𝐰)=π‘Ž(𝑒)πˆπ‘‘ξ€Έ,𝐰+(𝑏(𝑒)𝝈,𝐰),βˆ€π°βˆˆπ»(div,Ξ©),𝝈(π‘₯,0)=βˆ‡π‘’0𝝈(π‘₯),𝑑(π‘₯,0)=βˆ‡π‘’1𝑒(π‘₯),𝐩(π‘₯,0)=π‘Ž0ξ€Έβˆ‡π‘’1(𝑒π‘₯)+𝑏0ξ€Έβˆ‡π‘’0(π‘₯).(2.3)

In order to establish the equivalence between problem (2.2) and the weak form (2.3), we need the following technical lemmas.

Lemma 2.1 (see [25]). Let Ξ© be a bounded domain with a Lipschitz continuous boundary πœ•Ξ©. Then, for any 𝐩∈𝐻(div,Ξ©), there exists πœ™βˆˆπ»2⋂𝐻(Ξ©)10(Ξ©) and divergence free πœ“βˆˆπ»(div,Ξ©) such that βˆ‡β‹…πœ“=0 and 𝐩=βˆ‡πœ™+πœ“.

Lemma 2.2 (see [26]). Let Ξ© be a bounded domain with a Lipschitz continuous boundary πœ•Ξ©. Then, for any π‘”βˆˆπΏ2(Ξ©), there exists 𝐩∈(𝐻1(Ξ©))π‘‘βŠ‚π»(div,Ξ©) such that βˆ‡β‹…π©=𝑔.

Now we are in a position to state our main result in this subsection.

Theorem 2.3. Under the conditions of Lemmas 2.1 and 2.2, (𝑒,𝝈,𝐩)∈𝐻10(Ξ©)×𝐻(div,Ξ©)×𝐻(div,Ξ©) is a solution to the system (2.2) if and only if it is a solution to the weak form (2.3).

Proof. It is easy to check that any solution to the system (2.2) is a solution to the weak form (2.3). Hence, to prove the assertion, we only need to show that any solution to the weak form (2.3) is a solution to the system (2.2).
First, taking 𝐰=π©βˆ’π‘Ž(𝑒)πˆπ‘‘βˆ’π‘(𝑒)𝝈 in the third equation of (2.3) leads to ξ€·π©βˆ’π‘Ž(𝑒)πˆπ‘‘βˆ’π‘(𝑒)𝝈,π©βˆ’π‘Ž(𝑒)πˆπ‘‘ξ€Έβˆ’π‘(𝑒)𝝈=0,(2.4) which implies 𝐩=π‘Ž(𝑒)πˆπ‘‘βˆ’π‘(𝑒)𝝈.(2.5)
By Lemma 2.1, there exist πœ™βˆˆπ»2⋂𝐻(Ξ©)10(Ξ©) and divergence free πœ“βˆˆπ»(div,Ξ©) such that βˆ‡β‹…πœ“=0 and 𝝈=βˆ‡πœ™+πœ“. Choosing 𝝈=βˆ‡πœ™+πœ“ in the second equation of (2.3) yields (βˆ‡πœ™+πœ“,βˆ‡π‘£)=(βˆ‡π‘’,βˆ‡π‘£),βˆ€π‘£βˆˆπ»10(Ξ©).(2.6) By the divergence theorem [1], one has (πœ“,βˆ‡π‘£)=βˆ’(βˆ‡β‹…πœ“,𝑣)=0,βˆ€π‘£βˆˆπ»10(Ξ©).(2.7) Substituting (2.7) into (2.6) yields (βˆ‡πœ™,βˆ‡π‘£)=(βˆ‡π‘’,βˆ‡π‘£),βˆ€π‘£βˆˆπ»10(Ξ©),(2.8) which means that βˆ‡πœ™=βˆ‡π‘’,𝝈=βˆ‡π‘’+πœ“.(2.9) Inserting (2.5) and (2.9) into the first equation of (2.2) and applying the divergence theorem to the first term, for any πͺ∈𝐻(div,Ξ©), one has ξ€·π‘’π‘‘π‘‘ξ€Έβˆ’ξ€·πœ“,βˆ‡β‹…πͺπ‘‘π‘‘ξ€Έβˆ’ξ€·ξ€·π‘Žξ€·,πͺβˆ‡β‹…(𝑒)βˆ‡π‘’π‘‘+πœ“π‘‘ξ€Έ+ξ€Έξ€Έ,βˆ‡β‹…πͺ(βˆ‡β‹…(𝑏(𝑒)(βˆ‡π‘’+πœ“)),βˆ‡β‹…πͺ)=(𝑓,βˆ‡β‹…πͺ).(2.10) Instituting πͺ=πœ“π‘‘ into (2.10) and using βˆ‡β‹…πœ“π‘‘=0 lead to ξ€·πœ“0=𝑑𝑑,πœ“π‘‘ξ€Έ=12π‘‘ξ€·πœ“π‘‘π‘‘π‘‘,πœ“π‘‘ξ€Έ.(2.11) Integrating from 0 to 𝑑 with respect to time results in ξ€·πœ“π‘‘(π‘₯,𝑑),πœ“π‘‘ξ€Έ=ξ€·πœ“(π‘₯,𝑑)𝑑(π‘₯,0),πœ“π‘‘ξ€Έ.(π‘₯,0)(2.12) Differentiating (2.9) with respect to 𝑑, one obtains πˆπ‘‘=βˆ‡π‘’π‘‘+πœ“π‘‘.(2.13) By the fifth equation in (2.3), we deduce that πœ“π‘‘,(π‘₯,0)=0(2.14) which implies πœ“π‘‘(π‘₯,𝑑)=0.(2.15) Integrating the equation πœ“π‘‘(π‘₯,𝑑)=0 with respect to 𝑑 from 0 to 𝑑 gives πœ“(π‘₯,𝑑)=πœ“(π‘₯,0).(2.16) By (2.9) and the forth equation in (2.2), we deduce πœ“(π‘₯,𝑑)=0,(2.17) which leads to 𝝈=βˆ‡π‘’.(2.18) Therefore, (2.10) can equivalently be transformed into the following equation: ξ€·π‘’π‘‘π‘‘ξ€Έβˆ’ξ€·ξ€·π‘Ž,βˆ‡β‹…πͺβˆ‡β‹…(𝑒)βˆ‡π‘’π‘‘ξ€Έξ€Έ=+𝑏(𝑒)βˆ‡π‘’,βˆ‡β‹…πͺ(𝑓,βˆ‡β‹…πͺ),βˆ€πͺ∈𝐻(div,Ξ©).(2.19) For 𝑓,π‘’π‘‘π‘‘βˆˆπΏ2(Ξ©), by Lemma 2.2, there exists π…βˆˆπ»(div,Ξ©) such that βˆ‡β‹…π…=π‘’π‘‘π‘‘βˆ’π‘“. Thus, (2.19) reduces to (βˆ‡β‹…π©,βˆ‡β‹…πͺ)=(βˆ‡β‹…π…,βˆ‡β‹…πͺ),βˆ€πͺ∈𝐻(div,Ξ©).(2.20) Recalling Lemma 2.1, one concludes that βˆ‡β‹…π…=βˆ‡β‹…π©,(2.21) that is, π‘’π‘‘π‘‘βˆ’βˆ‡β‹…π©=𝑓.(2.22) Combining this with (2.5) and (2.18) results in the desired assertion, and this completes the proof.

2.2. Numerical Scheme

Let π‘‡β„Ž be a quasi-uniform family of subdivision of domain Ξ©; that is, Ξ©=βˆͺπΎβˆˆπ‘‡β„ŽπΎ with β„Ž = max {diam(𝐾)βˆΆπΎβˆˆπ‘‡β„Ž}, and let π‘‰β„Ž be the finite-dimensional subspaces of 𝐻10(Ξ©) defined by π‘‰β„Ž=ξ€½π‘£β„Žβˆˆπ»10(Ξ©);π‘£β„Žβˆ£πΎβˆˆπ‘ƒπ‘šξ€Ύ,(𝐾)(2.23) where π‘ƒπ‘š(𝐾) denotes the space of polynomials of degree at most π‘š on 𝐾. Moreover, we denote the vector space in mixed finite element spaces with index π‘˜ by π»β„Ž. It is well known that both π»β„Ž and π‘‰β„Ž satisfy the inverse property and the following approximation properties [26, 27]: infπ‘£β„Žβˆˆπ‘‰β„Žβ€–β€–π‘£βˆ’π‘£β„Žβ€–β€–β€–β€–+β„Žπ‘£βˆ’π‘£β„Žβ€–β€–1β‰€πΆβ„Žπ‘š+1β€–π‘£β€–π‘š+1,π‘£βˆˆπ»π‘š+1(Ξ©),infπͺβ„Žβˆˆπ‘Šβ„Žβ€–β€–πͺβˆ’πͺβ„Žβ€–β€–β‰€πΆβ„Žπ‘˜+1β€–πͺβ€–π‘˜+1,πͺβ„Žβˆˆξ€·π»π‘˜+1ξ€Έ(Ξ©)𝑑.(2.24)

Let Ξ β„ŽβˆΆπ»β†’π»β„Ž denote the Raviart-Thomas interpolation operator [28] which satisfies ξ€·ξ€·βˆ‡β‹…πͺβˆ’Ξ β„Žπͺξ€Έ,βˆ‡β‹…πͺβ„Žξ€Έ=0,βˆ€πͺβ„Žβˆˆπ»β„Ž,(2.25) and the following estimates [26, 28, 29] β€–β€–πͺβˆ’Ξ β„Žπͺβ€–β€–β‰€πΆβ„Žπ‘˜+1β€–πͺβ€–π‘˜+1,(2.26)β€–β€–ξ€·βˆ‡β‹…πͺβˆ’Ξ β„Žπͺξ€Έβ€–β€–β‰€πΆβ„Žπ‘˜β€–πͺβ€–π‘˜+1.(2.27)

With the above notations, the semidiscrete 𝐻1-Galerkin expanded mixed finite element method for system (2.3) is reduced to find a triple (π‘’β„Ž,πˆβ„Ž,π©β„Ž)βˆˆπ‘‰β„ŽΓ—π»β„ŽΓ—π»β„Ž such that ξ€·πˆβ„Žπ‘‘π‘‘,πͺβ„Žξ€Έ+ξ€·βˆ‡β‹…π©β„Ž,βˆ‡β‹…πͺβ„Žξ€Έξ€·=βˆ’π‘“,βˆ‡β‹…πͺβ„Žξ€Έ,βˆ€πͺβ„Žβˆˆπ»β„Ž,ξ€·πˆβ„Ž,βˆ‡π‘£β„Žξ€Έ=ξ€·βˆ‡π‘’β„Ž,βˆ‡π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‰β„Ž,ξ€·π©β„Ž,π°β„Žξ€Έ=ξ€·π‘Žξ€·π‘’β„Žξ€Έπˆβ„Žπ‘‘,π°β„Žξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπˆβ„Ž,π°β„Žξ€Έ,βˆ€π°β„Žβˆˆπ»β„Ž,π©β„Ž(π‘₯,0)=Ξ β„Žπˆπ©(π‘₯,0),β„Ž(π‘₯,0)=Ξ β„Žβˆ‡π‘’0(𝝈π‘₯),β„Žπ‘‘(π‘₯,0)=Ξ β„Žβˆ‡π‘’1(π‘₯).(2.28)

For the 𝐻1-Galerkin expanded mixed finite element scheme (2.28), we claim that there exists a unique solution.

In fact, set π‘‰β„Ž=span{πœ‘π‘–}𝑁𝑖=1 and π»β„Ž=span{πœ“π‘—}𝑀𝑗=1. Then πˆβ„Ž,π©β„Žβˆˆπ»β„Ž and π‘’β„Žβˆˆπ‘‰β„Ž, and; hence, πˆβ„Ž=𝑀𝑗=1𝑝𝑖(𝑑)πœ“π‘–(π‘₯),π©β„Ž=𝑀𝑗=1πœ†π‘–(𝑑)πœ“π‘–(π‘₯),π‘’β„Ž=𝑁𝑖=1𝑒𝑖(𝑑)πœ‘π‘–(π‘₯).(2.29) Taking πͺβ„Ž=πœ“π‘—, π°β„Ž=πœ“π‘—, 𝑗=1,2,…,𝑀, π‘£β„Ž=πœ‘π‘–, 𝑖=1,2,…,𝑁 in (2.28) leads to 𝐴𝑃𝑑𝑑+𝐡Λ=𝐹,π·π‘ˆ=𝐢𝑃,𝐴Λ=𝑀(π‘ˆ)𝑃𝑑+𝑁(π‘ˆ)𝑃,(2.30) whereξ€·πœ“π΄=𝑖(π‘₯),πœ“π‘—ξ€Έ(π‘₯)𝑀×𝑀𝑝,𝑃=1,𝑝2,…,𝑝𝑀𝑇,𝐡=βˆ‡β‹…πœ“π‘–(π‘₯),βˆ‡β‹…πœ“π‘—ξ€Έ(π‘₯)π‘€Γ—π‘€ξ€·πœ†,Ξ›=1,πœ†2,…,πœ†π‘€ξ€Έπ‘‡,𝐷=βˆ‡πœ‘π‘–(π‘₯),βˆ‡πœ‘π‘—ξ€Έ(π‘₯)𝑁×𝑁𝑒,π‘ˆ=1,𝑒2,…,𝑒𝑁𝑇,ξ€·πœ“πΆ=𝑖(π‘₯),βˆ‡πœ‘π‘—ξ€Έ(π‘₯)π‘Γ—π‘€ξ€·π‘Ž,𝑀(π‘ˆ)=(π‘ˆ)πœ“π‘–(π‘₯),πœ“π‘—ξ€Έ(π‘₯)𝑀×𝑀,𝑁𝑏(π‘ˆ)=(π‘ˆ)πœ“π‘–(π‘₯),πœ“π‘—ξ€Έ(π‘₯)𝑀×𝑀,𝐹=βˆ’π‘“,βˆ‡β‹…πœ“π‘—ξ€Έ(π‘₯)𝑀×1,(2.31) and 𝑃(0), 𝑃𝑑(0) are given.

Note that matrix 𝐴 in (2.31) is positive definite. Thus, by the third equation in (2.30), one has Ξ›=π΄βˆ’1𝑀𝑃𝑑+𝑁𝑃.(2.32) Inserting the above equality into the first equation of (2.30) yields 𝐴𝑃𝑑𝑑+π΅π΄βˆ’1𝑀𝑃𝑑+π΅π΄βˆ’1𝑁𝑃=𝐹.(2.33) By the standard arguments on the initial-value problem of a system of ordinary differential equations, we can obtain existence and uniqueness of 𝑃. The existence and uniqueness of π‘ˆ and Ξ› follow from the existence and uniqueness of 𝑃.

3. Error Analysis

This section is devoted to the error estimates for the 𝐻1-Galerkin expanded mixed finite element method.

For error analysis in the following, we need to introduce a projection operator. Let π‘…β„ŽβˆΆπ»10(Ξ©)β†’π‘‰β„Ž be the Ritz projection defined by ξ€·βˆ‡ξ€·π‘’βˆ’π‘…β„Žπ‘’ξ€Έ,βˆ‡π‘£β„Žξ€Έ=0,βˆ€π‘£β„Žβˆˆπ‘‰β„Ž.(3.1) Then the following approximation holds [27]: β€–β€–π‘’βˆ’π‘…β„Žπ‘’β€–β€–β€–β€–βˆ‡ξ€·+β„Žπ‘’βˆ’π‘…β„Žπ‘’ξ€Έβ€–β€–β‰€πΆβ„Žπ‘š+1β€–π‘’β€–π‘š+1.(3.2)

Let π©βˆ’π©β„Ž=ξ€·π©βˆ’Ξ β„Žπ©ξ€Έ+ξ€·Ξ β„Žπ©βˆ’π©β„Žξ€Έ=πœ‚+𝜁,πˆβˆ’πˆβ„Ž=ξ€·πˆβˆ’Ξ β„Žπˆξ€Έ+ξ€·Ξ β„Žπˆβˆ’πˆβ„Žξ€Έ=πœƒ+πœ‰,π‘’βˆ’π‘’β„Ž=ξ€·π‘’βˆ’π‘…β„Žπ‘’ξ€Έ+ξ€·π‘…β„Žπ‘’βˆ’π‘’β„Žξ€Έ=𝛼+𝛽.(3.3) Utilizing (2.3), (2.28), and auxiliary projections (3.1), (2.25), we can obtain the following error equations:ξ€·πœ‰π‘‘π‘‘,πͺβ„Žξ€Έ+ξ€·βˆ‡β‹…πœ,βˆ‡β‹…πͺβ„Žξ€Έξ€·πœƒ=βˆ’π‘‘,πͺβ„Žξ€Έ,βˆ€πͺβ„Žβˆˆπ»β„Ž,(3.4)ξ€·πœ‰,βˆ‡π‘£β„Žξ€Έ=ξ€·βˆ‡π›½,βˆ‡π‘£β„Žξ€Έβˆ’ξ€·πœƒ,βˆ‡π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘‰β„Ž,(3.5)ξ€·πœ,π°β„Žξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘,π°β„Žξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰,π°β„Žξ€Έ=ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,π°β„Žξ€Έ+ξ€·πˆξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,π°β„Žξ€Έ+ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘,π°β„Žξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒ,π°β„Žξ€Έβˆ’ξ€·πœ‚,π°β„Žξ€Έ,βˆ€π°β„Žβˆˆπ»β„Ž.(3.6)

Theorem 3.1. Let (𝑒,𝝈,𝐩) and (π‘’β„Ž,πˆβ„Ž,π©β„Ž) be the solutions to (2.3) and (2.28), respectively. Then the following error estimates hold: β€–β€–(π‘Ž)π‘’βˆ’π‘’β„Žβ€–β€–1β‰€πΆβ„Žmin(π‘˜+1,π‘š),β€–β€–ξ€·(𝑏)βˆ‡β‹…πˆβˆ’πˆβ„Žξ€Έβ€–β€–β‰€πΆβ„Žmin(π‘˜,π‘š+1),β€–β€–(𝑐)π‘’βˆ’π‘’β„Žβ€–β€–+β€–β€–πˆβˆ’πˆβ„Žβ€–β€–+β€–β€–π©βˆ’π©β„Žβ€–β€–β‰€πΆβ„Žmin(π‘˜+1,π‘š+1),(3.7) where π‘˜β‰₯1 and π‘šβ‰₯1 for 𝑑=2,3, and the positive constant 𝐢 depends on β€–π‘’π‘‘β€–πΏβˆž(π»π‘š+1), β€–π‘’β€–πΏβˆž(π»π‘š+1), β€–π©π‘‘β€–πΏβˆž(π»π‘˜+1), β€–π©β€–πΏβˆž(π»π‘˜+1), β€–πˆπ‘‘β€–πΏβˆž(π»π‘˜+1), β€–πˆπ‘‘π‘‘β€–πΏβˆž(π»π‘˜+1), β€–πˆβ€–πΏβˆž(π»π‘˜+1).

Proof. Since estimates of πœƒ, πœ‚, and 𝛼 can be obtained by (3.2) and (2.26), it suffices to estimate πœ‰, 𝜁, and 𝛽.
Instituting π°β„Ž=πœ‰π‘‘π‘‘ into (3.6) and πͺβ„Ž=𝜁 in (3.4) gives ξ€·π‘Žξ€·π‘’(βˆ‡β‹…πœ,βˆ‡β‹…πœ)+β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰,πœ‰π‘‘π‘‘ξ€Έξ€·πˆ=βˆ’π‘‘ξ€·π‘Žξ€·π‘’(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆξ€·π‘ξ€·π‘’(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒ,πœ‰π‘‘π‘‘ξ€Έ+ξ€·πœ‚,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πœƒπ‘‘π‘‘ξ€Έ.,𝜁(3.8) It is easy to check that ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘π‘‘,πœ‰π‘‘ξ€Έ=12π‘‘ξ€·π‘Žξ€·π‘’π‘‘π‘‘β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έβˆ’12ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘ξ€Έ,ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰,πœ‰π‘‘π‘‘ξ€Έ=π‘‘ξ€·π‘ξ€·π‘’π‘‘π‘‘β„Žξ€Έπœ‰,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έ,ξ€·πœ‚,πœ‰π‘‘π‘‘ξ€Έ=π‘‘ξ€·π‘‘π‘‘πœ‚,πœ‰π‘‘ξ€Έβˆ’ξ€·πœ‚π‘‘,πœ‰π‘‘ξ€Έ,ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έ=π‘‘ξ€·πˆπ‘‘π‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έ,ξ€·πˆξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έ=π‘‘ξ€·πˆξ€·ξ€·π‘’π‘‘π‘‘π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έβˆ’ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έ,ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έ=π‘‘ξ€·π‘Žξ€·π‘’π‘‘π‘‘β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,πœ‰π‘‘ξ€Έ,ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒ,πœ‰π‘‘π‘‘ξ€Έ=π‘‘ξ€·π‘ξ€·π‘’π‘‘π‘‘β„Žξ€Έπœƒ,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έ.(3.9) Thus, (3.8) can be written as 1(βˆ‡β‹…πœ,βˆ‡β‹…πœ)+2π‘‘ξ€·π‘Žξ€·π‘’π‘‘π‘‘β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έ+π‘‘ξ€·π‘ξ€·π‘’π‘‘π‘‘β„Žξ€Έπœ‰,πœ‰π‘‘ξ€Έ=12ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘ξ€Έ+ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έβˆ’ξ€·πœƒπ‘‘π‘‘ξ€Έ+𝑑,πœξ€·π‘‘π‘‘πœ‚,πœ‰π‘‘ξ€Έβˆ’ξ€·πœ‚π‘‘,πœ‰π‘‘ξ€Έβˆ’π‘‘ξ€·πˆπ‘‘π‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έ+ξ€·πˆπ‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έ+ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έβˆ’π‘‘ξ€·πˆξ€·ξ€·π‘’π‘‘π‘‘π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έ+ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έ+ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έβˆ’π‘‘ξ€·π‘Žξ€·π‘’π‘‘π‘‘β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έ+ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘ξ€Έ+ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,πœ‰π‘‘ξ€Έβˆ’π‘‘ξ€·π‘ξ€·π‘’π‘‘π‘‘β„Žξ€Έπœƒ,πœ‰π‘‘ξ€Έ+ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έ.(3.10) Integrating this system from 0 to 𝑑 yields ξ€œπ‘‘0β€–βˆ‡β‹…πœβ€–21π‘‘πœ+2ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰,πœ‰π‘‘ξ€Έ=ξ€·πœ‚,πœ‰π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έβˆ’ξ€·πˆξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒ,πœ‰π‘‘ξ€Έ+12ξ€œπ‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έβˆ’ξ€œπ‘‘πœπ‘‘0ξ€·πœƒπ‘‘π‘‘ξ€Έξ€œ,πœπ‘‘πœβˆ’π‘‘0ξ€·πœ‚π‘‘,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·πˆπ‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έ+ξ€œπ‘‘πœπ‘‘0ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·πˆπ‘‘ξ€·π‘ξ€·π‘’(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘ξ€Έ+ξ€œπ‘‘πœπ‘‘0ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘ξ€Έ+ξ€œπ‘‘πœπ‘‘0ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έπ‘‘πœ.(3.11) In what follows, we, respectively, analyze the terms on the right-hand side of (3.11). By the Cauchy-Schwartz inequality, we can bound the sixth term on the right-hand side of (3.11) as follows: ||||ξ€œπ‘‘012ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘ξ€Έ||||=1π‘‘πœ2||||ξ€œπ‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’π‘‘πœ‰π‘‘,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑012ξ€·π‘Žπ‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·β„Žπ‘‘βˆ’π‘’π‘‘ξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έ||||ξ€œπ‘‘πœβ‰€πΆπ‘‘0β€–β€–πœ‰π‘‘β€–β€–2β€–β€–πœ‰π‘‘πœ+πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚π‘‘πœ.(3.12) For the seventh term on the right-hand side of (3.11), one has ||||ξ€œπ‘‘0ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘ξ€Έ||||=||||ξ€œπ‘‘πœπ‘‘0ξ€·π‘π‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·β„Žπ‘‘βˆ’π‘’π‘‘ξ€Έπœ‰,πœ‰π‘‘ξ€Έ+ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’π‘‘πœ‰,πœ‰π‘‘ξ€Έ||||ξ€œπ‘‘πœβ‰€πΆπ‘‘0ξ‚€β€–β€–πœ‰π‘‘β€–β€–2+β€–πœ‰β€–2ξ‚β€–β€–πœ‰π‘‘πœ+πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–πœ‰β€–2ξ‚π‘‘πœ.(3.13) For the term βˆ«π‘‘0(πˆπ‘‘(π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’(π‘’β„Ž)π‘’β„Žπ‘‘),πœ‰π‘‘)π‘‘πœ on the right side of (3.11), we have ||||ξ€œπ‘‘0ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έ||||=||||ξ€œπ‘‘πœπ‘‘0ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)βˆ’π‘Žπ‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€Έπ‘‘+π‘Žπ‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·π‘‘βˆ’π‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έ||||ξ€œπ‘‘πœβ‰€πΆπ‘‘0‖𝛼‖2+‖𝛽‖2+‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚π‘‘πœ.(3.14) Similarly, ||||ξ€œπ‘‘0ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έ||||β‰€ξ€œπ‘‘πœπ‘‘0||ξ€·πˆξ€·π‘π‘’(𝑒)βˆ’π‘π‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€Έπ‘‘+π‘π‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·π‘‘βˆ’π‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘ξ€Έ||ξ€œπ‘‘πœβ‰€πΆπ‘‘0‖𝛼‖2+‖𝛽‖2+‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœ‰π‘‘β€–β€–2||||ξ€œπ‘‘πœ,𝑑0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘ξ€Έ||||=||||ξ€œπ‘‘πœπ‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·β„Žπ‘‘βˆ’π‘’π‘‘ξ€Έπœƒπ‘‘,πœ‰π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’π‘‘πœƒπ‘‘,πœ‰π‘‘ξ€Έ||||β€–β€–πœ‰π‘‘πœβ‰€πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœƒπ‘‘β€–β€–2ξ‚ξ€œπ‘‘πœ+𝐢𝑑0ξ‚€β€–β€–πœƒπ‘‘β€–β€–2+β€–β€–πœ‰π‘‘β€–β€–2||||ξ€œπ‘‘πœ,𝑑0ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘ξ€Έ||||β‰€ξ€œπ‘‘πœπ‘‘0||ξ€·π‘π‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·β„Žπ‘‘βˆ’π‘’π‘‘ξ€Έπœƒ,πœ‰π‘‘ξ€Έ||ξ€œπ‘‘πœ+𝑑0||ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’π‘‘πœƒ,πœ‰π‘‘ξ€Έ||β€–β€–πœ‰π‘‘πœβ‰€πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–πœƒβ€–2ξ‚ξ€œπ‘‘πœ+𝐢𝑑0ξ‚€β€–πœƒβ€–2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚π‘‘πœ.(3.15)
Inserting (3.12)–(3.15) into (3.11) and using the Cauchy-Schwartz inequality lead to ξ€œπ‘‘0β€–βˆ‡β‹…πœβ€–21π‘‘πœ+2ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰,πœ‰π‘‘ξ€Έξ‚€β‰€πΆβ€–πœ‚β€–2+β€–β€–πœ‰π‘‘β€–β€–2+‖𝛼‖2+‖𝛽‖2+β€–πœƒβ€–2+β€–β€–πœƒπ‘‘β€–β€–2ξ‚ξ€œ+𝐢𝑑0ξ‚€β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–β€–πœ‰π‘‘β€–β€–2+β€–β€–πœ‚π‘‘β€–β€–2+‖𝛼‖2+‖𝛽‖2+β€–πœβ€–2ξ‚β€–β€–πœ‰π‘‘πœ+πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚β€–β€–πœ‰π‘‘πœ+πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0ξ‚€β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+β€–πœ‰β€–2ξ‚ξ€œπ‘‘πœ+𝐢𝑑0ξ‚€β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+β€–πœ‰β€–2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚π‘‘πœ.(3.16) Integrating (3.16) from 0 to 𝑑, using the fact (𝑏(π‘’β„Ž)πœ‰,πœ‰π‘‘)=(1/2)(𝑑/𝑑𝑑)(𝑏(π‘’β„Ž)πœ‰,πœ‰)βˆ’(1/2)(𝑏𝑒(π‘’β„Ž)π‘’β„Žπ‘‘πœ‰,πœ‰π‘‘) and the inequality ξ€œπ‘‘0ξ€œπœ0||||πœ“(𝑠)2ξ€œπ‘‘π‘ π‘‘πœβ‰€πΆπ‘‘0||||πœ“(𝑠)2𝑑𝑠,(3.17) yields β€–πœ‰β€–2β€–β€–πœ‰β‰€πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚β€–β€–πœ‰π‘‘πœ+πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0ξ‚€β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+β€–πœ‰β€–2ξ‚ξ€œπ‘‘πœ+𝐢𝑑0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2+‖𝛼‖2+‖𝛽‖2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœ‚π‘‘β€–β€–2+β€–πœ‰β€–2+β€–β€–πœ‰π‘‘β€–β€–2+β€–πœβ€–2ξ‚π‘‘πœ.(3.18) Thus, to estimate β€–πœ‰β€–, we need to estimate ‖𝛽‖, ‖𝛽𝑑‖, β€–πœβ€–, and β€–πœ‰π‘‘β€–. Taking π‘£β„Ž=𝛽 in (3.5) leads to (βˆ‡π›½,βˆ‡π›½)=(πœ‰,βˆ‡π›½)+(πœƒ,βˆ‡π›½).(3.19) By the Cauchy-Schwartz inequality, we obtain β€–βˆ‡π›½β€–β‰€πΆ(β€–πœ‰β€–+β€–πœƒβ€–).(3.20) Note that π›½βˆˆπ‘‰β„ŽβŠ‚π»10(Ξ©) and β€–π›½β€–β‰€πΆβ€–βˆ‡π›½β€–. We further have ‖𝛽‖≀𝐢(β€–πœ‰β€–+β€–πœƒβ€–).(3.21) Differentiating (3.5) with respect to 𝑑 and choosing π‘£β„Ž=𝛽𝑑 gives β€–β€–βˆ‡π›½π‘‘β€–β€–ξ€·β€–β€–πœ‰β‰€πΆπ‘‘β€–β€–+β€–β€–πœƒπ‘‘β€–β€–ξ€Έ.(3.22) Similarly, since π›½βˆˆπ‘‰β„ŽβŠ‚π»10(Ξ©), one has β€–π›½π‘‘β€–β‰€β€–βˆ‡π›½π‘‘β€–β‰€πΆ(β€–πœ‰π‘‘β€–+β€–πœƒπ‘‘β€–).
Taking π°β„Ž=𝜁 in (3.6), one has ξ€·π‘Žξ€·π‘’(𝜁,𝜁)=β„Žξ€Έπœ‰π‘‘ξ€Έ+𝑏𝑒,πœβ„Žξ€Έξ€Έ+ξ€·πˆπœ‰,πœπ‘‘ξ€·π‘Žξ€·π‘’(𝑒)βˆ’π‘Žβ„Žξ€Έ+ξ€·πˆξ€·ξ€·π‘’ξ€Έξ€Έ,πœπ‘(𝑒)βˆ’π‘β„Žξ€Έ+ξ€·π‘Žξ€·π‘’ξ€Έξ€Έ,πœβ„Žξ€Έπœƒπ‘‘ξ€Έ+𝑏𝑒,πœβ„Žξ€Έξ€Έπœƒ,πœβˆ’(πœ‚,𝜁).(3.23) By the Cauchy-Schwartz inequality, we obtain ξ€·β€–β€–πœƒβ€–πœβ€–β‰€πΆβ€–πœ‰β€–+π‘‘β€–β€–β€–β€–πœ‰+β€–πœƒβ€–+β€–πœ‚β€–+‖𝛼‖+‖𝛽‖+𝑑‖‖.(3.24) To bound β€–πœ‰π‘‘β€–2, we differentiate (3.6) with respect to 𝑑 to obtain ξ€·πœπ‘‘,π°β„Žξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘π‘‘,π°β„Žξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,π°β„Žξ€Έ=ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,π°β„Žξ€Έ+ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,π°β„Žξ€Έ+ξ€·πˆπ‘‘π‘‘ξ€·π‘Žξ€·π‘’(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,π°β„Žξ€Έ+ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,π°β„Žξ€Έ+ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,π°β„Žξ€Έ+ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,π°β„Žξ€Έ+ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,π°β„Žξ€Έ+ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,π°β„Žξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒπ‘‘,π°β„Žξ€Έ+ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,π°β„Žξ€Έβˆ’ξ€·πœ‚π‘‘,π°β„Žξ€Έ,βˆ€π°β„Žβˆˆπ»β„Ž.(3.25) Testing (3.25) with π°β„Ž=πœ‰π‘‘π‘‘ and (3.4) with πͺβ„Ž=πœπ‘‘ and combining the resulting equations together lead to ξ€·βˆ‡β‹…πœ,βˆ‡β‹…πœπ‘‘ξ€Έ+ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘π‘‘,πœ‰π‘‘π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έξ€·π‘Ž=βˆ’π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘π‘‘ξ€·π‘Žξ€·π‘’(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πœƒπ‘‘,πœπ‘‘ξ€Έβˆ’ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘π‘‘ξ€Έ+ξ€·πœ‚π‘‘,πœ‰π‘‘π‘‘ξ€Έ.(3.26) Note that ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έ=12π‘‘ξ€·π‘ξ€·π‘’π‘‘π‘‘β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έβˆ’12ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘ξ€Έ,ξ€·βˆ‡β‹…πœ,βˆ‡β‹…πœπ‘‘ξ€Έ=12π‘‘ξ€·πœƒπ‘‘π‘‘(βˆ‡β‹…πœ,βˆ‡β‹…πœ),𝑑,πœπ‘‘ξ€Έ=π‘‘ξ€·πœƒπ‘‘π‘‘π‘‘ξ€Έβˆ’ξ€·πœƒ,πœπ‘‘π‘‘ξ€Έ.,𝜁(3.27) Thus, (3.26) can be rewritten as 12π‘‘ξ€·π‘Žξ€·π‘’π‘‘π‘‘(βˆ‡β‹…πœ,βˆ‡β‹…πœ)+β„Žξ€Έπœ‰π‘‘π‘‘,πœ‰π‘‘π‘‘ξ€Έ+12π‘‘ξ€·π‘ξ€·π‘’π‘‘π‘‘β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έξ€·π‘Ž=βˆ’π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’12ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆπ‘‘ξ€·π‘ξ€·π‘’(𝑒)βˆ’π‘β„Žξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’π‘‘ξ€·πœƒπ‘‘π‘‘π‘‘ξ€Έ+ξ€·πœƒ,πœπ‘‘π‘‘ξ€Έβˆ’ξ€·π‘ξ€·π‘’,πœβ„Žξ€Έπœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘π‘‘+ξ€·πœ‚ξ€Έξ€Έπ‘‘,πœ‰π‘‘π‘‘ξ€Έ.(3.28) Integrating (3.28) from 0 to 𝑑 yields (ξ€œβˆ‡β‹…πœ,βˆ‡β‹…πœ)+𝑑0ξ€·π‘Žξ€·π‘’β„Žξ€Έπœ‰π‘‘π‘‘,πœ‰π‘‘π‘‘ξ€Έ+ξ€·π‘ξ€·π‘’β„Žξ€Έπœ‰π‘‘,πœ‰π‘‘ξ€Έξ€œ=βˆ’π‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έ1π‘‘πœβˆ’2ξ€œπ‘‘0ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€œπ‘‘πœπ‘‘0ξ€·πˆπ‘‘π‘‘ξ€·ξ€·π‘’π‘Ž(𝑒)βˆ’π‘Žβ„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έξ€œπ‘‘πœβˆ’π‘‘0ξ€·πˆπ‘‘ξ€·π‘Žπ‘’(𝑒)π‘’π‘‘βˆ’π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€œπ‘‘πœπ‘‘0ξ€·πˆπ‘‘ξ€·ξ€·π‘’π‘(𝑒)βˆ’π‘β„Žξ€Έξ€Έ,πœ‰π‘‘π‘‘ξ€Έξ€œπ‘‘πœβˆ’π‘‘0ξ€·πˆξ€·π‘π‘’(𝑒)π‘’π‘‘βˆ’π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘ξ€Έ,πœ‰π‘‘π‘‘ξ€Έβˆ’ξ€œπ‘‘πœπ‘‘0ξ€·π‘Žξ€·π‘’β„Žξ€Έπœƒπ‘‘π‘‘,πœ‰π‘‘π‘‘ξ€Έξ€œπ‘‘πœβˆ’π‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έξ€·πœƒπ‘‘πœβˆ’π‘‘ξ€Έ+ξ€œ,πœπ‘‘0ξ€·πœƒπ‘‘π‘‘ξ€Έβˆ’ξ€œ,πœπ‘‘πœπ‘‘0ξ€·π‘ξ€·π‘’β„Žξ€Έπœƒπ‘‘,πœ‰π‘‘π‘‘ξ€Έξ€œπ‘‘πœβˆ’π‘‘0ξ€·π‘π‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœƒ,πœ‰π‘‘π‘‘ξ€Έξ€œπ‘‘πœ+𝑑0ξ€·πœ‚π‘‘,πœ‰π‘‘π‘‘ξ€Έπ‘‘πœ.(3.29)
For the first term on the right-hand side of (3.29), by the Cauchy-Schwarz inequality and Young's inequality, for sufficiently small constant πœ€>0, it holds that ||||βˆ’ξ€œπ‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’β„Žπ‘‘πœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έ||||≀||||ξ€œπ‘‘πœπ‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žπ‘’ξ€Έξ€·β„Žπ‘‘βˆ’π‘’π‘‘ξ€Έπœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έ||||+||||ξ€œπ‘‘πœπ‘‘0ξ€·π‘Žπ‘’ξ€·π‘’β„Žξ€Έπ‘’π‘‘πœ‰π‘‘,πœ‰π‘‘π‘‘ξ€Έ||||β€–β€–πœ‰π‘‘πœβ‰€πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2ξ‚ξ€œπ‘‘πœ+𝐢𝑑0β€–β€–πœ‰π‘‘β€–β€–2ξ€·β€–β€–πœ‰π‘‘πœ+πœ€1+π‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€Έξ€œπ‘‘0β€–β€–πœ‰π‘‘π‘‘β€–β€–2π‘‘πœ.(3.30) Similarly, we can bound (3.29) as follows: β€–βˆ‡β‹…πœβ€–2+β€–β€–πœ‰π‘‘β€–β€–2+ξ€œπ‘‘0β€–β€–πœ‰π‘‘π‘‘β€–β€–2β€–β€–πœ‰π‘‘πœβ‰€πΆπ‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2ξ‚ξ€·β€–β€–πœ‰π‘‘πœ+πœ€1+π‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)ξ€Έξ€œπ‘‘0β€–β€–πœ‰π‘‘π‘‘β€–β€–2π‘‘πœ+πΆβ€–πœ‰β€–πΏβˆž(0,𝑑;𝐿∞)ξ€œπ‘‘0‖‖𝛼𝑑‖‖2+‖‖𝛽𝑑‖‖2ξ‚ξ€·π‘‘πœ+πœ€1+β€–πœ‰β€–πΏβˆž(0,𝑑;𝐿∞)ξ€Έξ€œπ‘‘0β€–β€–πœ‰π‘‘π‘‘β€–β€–2ξ€œπ‘‘πœ+𝐢𝑑0ξ‚€β€–πœ‰β€–2+β€–πœβ€–2+β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+‖‖𝛼𝑑‖‖2+‖𝛽‖2+‖‖𝛽𝑑‖‖2+β€–β€–πœ‚π‘‘β€–β€–2+β€–β€–πœ‰π‘‘β€–β€–2+β€–β€–πœƒπ‘‘πœπ‘‘β€–β€–β€–πœβ€–.(3.31) In the following error analysis, we make an induction hypothesis: ξ€·β€–β€–πœ‰π‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)+β€–πœ‰β€–πΏβˆž(0,𝑑;𝐿∞)≀1.(3.32) Utilizing (3.32), (3.24), (3.22), (3.21), and Young's inequality, one can reduce (3.31) to β€–βˆ‡β‹…πœβ€–2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚€β€–β‰€πΆπœ‰β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+β€–πœ‚β€–2ξ‚ξ€œ+𝐢𝑑0ξ‚€β€–πœ‰β€–2+β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+‖‖𝛼tβ€–β€–2+β€–β€–πœ‚π‘‘β€–β€–2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚π‘‘πœ.(3.33) Then by Gronwall's inequality, we obtain β€–βˆ‡β‹…πœβ€–2+β€–β€–πœ‰π‘‘β€–β€–2ξ‚€β€–β‰€πΆπœ‰β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+β€–πœ‚β€–2ξ‚ξ€œ+𝐢𝑑0ξ‚€β€–πœ‰β€–2+β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+‖‖𝛼𝑑‖‖2+β€–β€–πœ‚π‘‘β€–β€–2ξ‚π‘‘πœ.(3.34) Furthermore, by (3.24) and (3.34), one has β€–πœβ€–2ξ‚€β€–β‰€πΆπœ‰β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+β€–πœ‚β€–2ξ‚ξ€œ+𝐢𝑑0ξ‚€β€–πœ‰β€–2+β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+‖‖𝛼𝑑‖‖2+β€–β€–πœ‚π‘‘β€–β€–2ξ‚π‘‘πœ.(3.35) Therefore, by the estimates of ‖𝛽‖, ‖𝛽𝑑‖, β€–πœβ€–, and β€–πœ‰π‘‘β€–, it follows that β€–πœ‰β€–2ξ€œβ‰€πΆπ‘‘0ξ‚€β€–πœ‰β€–2+β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+‖‖𝛼𝑑‖‖2+β€–πœ‚β€–2+β€–β€–πœ‚π‘‘β€–β€–2ξ‚π‘‘πœ.(3.36) Applying Gronwall's inequality to the above equation and using the estimates of projection operators give β€–πœ‰β€–2ξ€œβ‰€πΆπ‘‘0ξ‚€β€–β€–πœƒπ‘‘π‘‘β€–β€–2+β€–β€–πœƒπ‘‘β€–β€–2+β€–πœƒβ€–2+‖𝛼‖2+‖‖𝛼𝑑‖‖2+β€–πœ‚β€–2+β€–β€–πœ‚π‘‘β€–β€–2ξ‚π‘‘πœβ‰€πΆβ„Žmin(2π‘˜+2,2π‘š+2)‖‖𝑒𝑑‖‖2πΏβˆžξ€·π»π‘š+1ξ€Έ+‖𝑒‖2πΏβˆžξ€·π»π‘š+1ξ€Έ+‖‖𝐩𝑑‖‖2πΏβˆžξ€·π»π‘˜+1ξ€Έ+‖𝐩‖2𝐿∞(π»π‘˜+1)+β€–β€–πˆπ‘‘β€–β€–2𝐿∞(π»π‘˜+1)+β€–β€–πˆπ‘‘π‘‘β€–β€–2𝐿∞(π»π‘˜+1)+β€–πˆβ€–2πΏβˆžξ€·π»π‘˜+1.(3.37) Inserting the estimate of β€–πœ‰β€– into (3.34) yields β€–β€–πœ‰π‘‘β€–β€–2β‰€πΆβ„Žmin(2π‘˜+2,2π‘š+2).(3.38) Thus, the estimates of 𝛽 and 𝜁 follow from the estimate of πœ‰.
Finally, according to the proof of the induction hypothesis in [23, 30], we can prove that the inductive hypothesis (3.32) holds. In fact, when 𝑑=0, then πœ‰(0)=0, πœ‰π‘‘(0)=0. Note that β€–πœ‰β€–πΏβˆž(0,𝑑;𝐿∞)+β€–πœ‰π‘‘β€–πΏβˆž(0,𝑑;𝐿∞) is continuous w.r.t. 𝑑. Then, we conclude that there exists 𝑑1∈(0,𝑇] such that β€–πœ‰β€–πΏβˆž(0,𝑑1;𝐿∞)+β€–β€–πœ‰π‘‘β€–β€–πΏβˆž(0,𝑑1;𝐿∞)≀1.(3.39) Set π‘‘βˆ—=sup𝑑1. Thus, β€–πœ‰β€–πΏβˆž(0,π‘‘βˆ—;𝐿∞)+β€–πœ‰π‘‘β€–πΏβˆž(0,π‘‘βˆ—;𝐿∞)≀1. Therefore, we have β€–β€–πœ‰ξ€·π‘‘βˆ—ξ€Έβ€–β€–+β€–β€–πœ‰π‘‘ξ€·π‘‘βˆ—ξ€Έβ€–β€–β‰€πΆβ„Žmin(π‘˜+1,π‘š+1).(3.40)
By inverse estimates, we deduce that, for any 0β‰€π‘‘β‰€π‘‘βˆ—, it holds that β€–πœ‰β€–πΏβˆž(0,𝑑;𝐿∞)+β€–β€–πœ‰π‘‘β€–β€–πΏβˆž(0,𝑑;𝐿∞)β‰€πΆβ„Žmin(π‘˜+1,π‘š+1)βˆ’π‘‘/2.(3.41) Then we can take β„Ž>0 sufficiently small such that β€–πœ‰β€–πΏβˆž(0,π‘‘βˆ—;𝐿∞)+β€–β€–πœ‰π‘‘β€–β€–πΏβˆž(0,π‘‘βˆ—;𝐿∞)<1.(3.42) Again, by the continuity of β€–πœ‰β€–πΏβˆž(0,𝑑;𝐿∞)+β€–πœ‰π‘‘β€–πΏβˆž(0,𝑑;𝐿∞), we conclude that there exists a positive constant 𝛿 such that β€–πœ‰β€–πΏβˆž(0,π‘‘βˆ—+𝛿;𝐿∞)+β€–β€–πœ‰π‘‘β€–β€–πΏβˆž(0,π‘‘βˆ—+𝛿;𝐿∞)≀1,(3.43) which contracts to the definition of π‘‘βˆ—. This completes the proof of the induction hypothesis.
Combining (3.21), (3.37), (3.2), (2.26), (2.27) with the estimates of auxiliary projections and utilizing the triangle inequality, we can derive the desired result.

Remark 3.2. By Theorem 3.1 and the standard embedding theorem, we can obtain the 𝐿∞ estimate for 𝑑=1 and 2 as follows: β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–πΏβˆž(𝐿∞)≀𝐢2||||lnβ„Žπ‘‘βˆ’1β„Žmin(π‘˜+1,π‘š+1).(3.44)

4. Conclusion

In this paper, 𝐻1-Galerkin mixed finite element method combining with expanded mixed element method is discussed for nonlinear viscoelasticity equations. This method solves the scalar unknown, its gradient, and its flux, directly. It is suitable for the case that the coefficient of the differential is a small tensor and does not need to be inverted. Furthermore, the formulation permits the use of standard continuous and piecewise (linear and higher-order) polynomials in contrast to continuously differentiable piecewise polynomials required by the standard 𝐻1-Galerkin methods and is free of the LBB condition which is required by the mixed finite element methods.

There are also some important issues to be addressed in the area; for example, one can consider numerical implementation and mathematical and numerical analysis of the full discrete procedure. This is an important and challenging topic in the future research.

Acknowledgments

This project is supported by the Natural Science Foundation of China (Grant no. 11171180, 10901096), the Shandong Provincial Natural Science Foundation (Grant no. ZR2009AL019), the Shandong Provincial Higher Educational Science and Technology Program (Grant no. J09LA53), and the Shandong Provincial Young Scientist Foundation (Grant no. 2008BS01008).

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