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

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: 𝑢_𝑡𝑡−∇⋅𝑎(𝑥,𝑢)𝑢∇(𝑢𝑢𝑥(_𝑡,𝑥+𝑡,)0𝑏=)(=𝑥0,,𝑢𝑢₀)(∇𝑥,𝑢𝑡)=∈𝑓𝜕Ω(𝑥×,𝑡𝐽),,(𝑥,𝑡)∈Ω×𝐽,𝑢(𝑥),𝑥∈Ω,_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω,(1.1) where Ω is a convex polygonal domain in 𝑅² with the Lipschitz continuous boundary 𝜕Ω, 𝐽=(0,𝑇] is the time interval with 0<𝑇<∞, and 𝑢₀(𝑥) and 𝑢₁(𝑥) are, respectively, the initial data functions defined on Ω. 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 𝑢₀(𝑥), 𝑢₁(𝑥), 𝑎(𝑢), 𝑏(𝑢), 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 𝐿_𝑝 (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 ∇𝑢 by the unknown scalar 𝑢 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 𝑢 and its flux ∇𝑢 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,Ω)={𝐯∈(𝐿²(Ω))^𝑑∶∇⋅𝐯∈𝐿²(Ω)} with norm ‖𝐯‖_𝐇(div;Ω)=(‖𝐯‖²+‖∇⋅𝐯‖²)^1/2 and 𝐻¹⁰(Ω)={𝑤∈𝐻¹(Ω)∶𝑤=0on𝜕Ω}. 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<𝑎₁≤𝑎(𝑥,𝑢),𝑏(𝑥,𝑢)≤𝑎₂.
(2) The functions 𝑎(𝑥,𝑢),𝑏(𝑥,𝑢),𝑎_𝑢(𝑥,𝑢), and 𝑏_𝑢(𝑥,𝑢) are Lipschitz continuous with respect to 𝑢, and there exists 𝐶₁>0 such that |𝜕𝑎/𝜕𝑢|+|𝜕𝑏/𝜕𝑢|+|𝜕²𝑎/𝜕𝑢²|+|𝜕²𝑏/𝜕𝑢²|≤𝐶₁.

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=𝑎(𝑥,𝑢)∇𝑢_𝑡+𝑏(𝑥,𝑢)∇𝑢,𝝈=∇𝑢,(2.1) and split (1.1) into a first-order system as follows: 𝐩=𝑢𝑎_𝑡𝑡−(𝝈𝑢)∇=𝝈⋅_𝑡∇𝐩𝑢=,𝑓,𝝈(𝑥,0)=∇+𝑢𝑏₀(𝑢)𝝈,𝝈(𝑥),_𝑡𝐩(𝑥,0)=𝑎𝑢(₀𝑥,0)=∇𝑢₁(𝑥),∇𝑢₁(𝑥)+𝑏𝑢₀∇𝑢₀(𝑥).(2.2) Then by Green's formula we can further define the following weak formulation of problem (2.2): find (𝑢,𝝈,𝐩)∈𝐻¹⁰(Ω)×𝐻(div,Ω)×𝐻(div,Ω) such that 𝝈_𝑡𝑡,𝐪+((∇𝝈,⋅∇𝐩,𝑣∇)=⋅𝐪()∇=𝑢,−∇𝑣(𝑓),,∇∀⋅𝑣𝐪)∈,𝐻∀₁₀𝐪∈𝐻(div,Ω),(𝐩,𝐰)=𝑎(𝑢)𝝈_𝑡(Ω),,𝐰𝝈(𝑥+,(0𝑏)(=𝑢)∇𝝈𝑢,₀𝐰),∀𝐰∈𝐻(div,Ω),𝝈(𝑥),_𝑡𝐩(𝑥,0)=𝑎𝑢(₀𝑥,0)=∇𝑢₁(𝑥),∇𝑢₁(𝑥)+𝑏𝑢₀∇𝑢₀(𝑥).(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 Ω be a bounded domain with a Lipschitz continuous boundary 𝜕Ω. Then, for any 𝐩∈𝐻(div,Ω), there exists 𝜙∈𝐻²(Ω)⋂𝐻¹⁰(Ω) and divergence free 𝜓∈𝐻(div,Ω) such that ∇⋅𝜓=0 and 𝐩=∇𝜙+𝜓.

Lemma 2.2 (see [26]). Let Ω be a bounded domain with a Lipschitz continuous boundary 𝜕Ω. Then, for any 𝑔∈𝐿²(Ω), there exists 𝐩∈(𝐻¹(Ω))^𝑑⊂𝐻(div,Ω) such that ∇⋅𝐩=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, (𝑢,𝝈,𝐩)∈𝐻¹⁰(Ω)×𝐻(div,Ω)×𝐻(div,Ω) 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=𝐩−𝑎(𝑢)𝝈_𝑡−𝑏(𝑢)𝝈 in the third equation of (2.3) leads to 𝐩−𝑎(𝑢)𝝈_𝑡−𝑏(𝑢)𝝈,𝐩−𝑎(𝑢)𝝈_𝑡−𝑏(𝑢)𝝈=0,(2.4) which implies 𝐩=𝑎(𝑢)𝝈_𝑡−𝑏(𝑢)𝝈.(2.5)
By Lemma 2.1, there exist 𝜙∈𝐻²(Ω)⋂𝐻¹⁰(Ω) and divergence free 𝜓∈𝐻(div,Ω) such that ∇⋅𝜓=0 and 𝝈=∇𝜙+𝜓. Choosing 𝝈=∇𝜙+𝜓 in the second equation of (2.3) yields (∇𝜙+𝜓,∇𝑣)=(∇𝑢,∇𝑣),∀𝑣∈𝐻¹⁰(Ω).(2.6) By the divergence theorem [1], one has (𝜓,∇𝑣)=−(∇⋅𝜓,𝑣)=0,∀𝑣∈𝐻¹⁰(Ω).(2.7) Substituting (2.7) into (2.6) yields (∇𝜙,∇𝑣)=(∇𝑢,∇𝑣),∀𝑣∈𝐻¹⁰(Ω),(2.8) which means that ∇𝜙=∇𝑢,𝝈=∇𝑢+𝜓.(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 𝐪∈𝐻(div,Ω), one has 𝑢_𝑡𝑡,∇⋅𝐪−𝜓_𝑡𝑡,𝐪−∇⋅𝑎(𝑢)∇𝑢_𝑡+𝜓_𝑡,∇⋅𝐪+(∇⋅(𝑏(𝑢)(∇𝑢+𝜓)),∇⋅𝐪)=(𝑓,∇⋅𝐪).(2.10) Instituting 𝐪=𝜓_𝑡 into (2.10) and using ∇⋅𝜓_𝑡=0 lead to 0=𝜓_𝑡𝑡,𝜓_𝑡=12𝑑𝑑𝑡𝜓^𝑡,𝜓_𝑡.(2.11) Integrating from 0 to 𝑡 with respect to time results in 𝜓_𝑡(𝑥,𝑡),𝜓_𝑡(𝑥,𝑡)=𝜓_𝑡(𝑥,0),𝜓_𝑡(𝑥,0).(2.12) Differentiating (2.9) with respect to 𝑡, one obtains 𝝈_𝑡=∇𝑢_𝑡+𝜓_𝑡.(2.13) By the fifth equation in (2.3), we deduce that 𝜓_𝑡(𝑥,0)=0,(2.14) which implies 𝜓_𝑡(𝑥,𝑡)=0.(2.15) Integrating the equation 𝜓_𝑡(𝑥,𝑡)=0 with respect to 𝑡 from 0 to 𝑡 gives 𝜓(𝑥,𝑡)=𝜓(𝑥,0).(2.16) By (2.9) and the forth equation in (2.2), we deduce 𝜓(𝑥,𝑡)=0,(2.17) which leads to 𝝈=∇𝑢.(2.18) Therefore, (2.10) can equivalently be transformed into the following equation: 𝑢_𝑡𝑡,∇⋅𝐪−∇⋅𝑎(𝑢)∇𝑢_𝑡+𝑏(𝑢)∇𝑢,∇⋅𝐪=(𝑓,∇⋅𝐪),∀𝐪∈𝐻(div,Ω).(2.19) For 𝑓,𝑢_𝑡𝑡∈𝐿²(Ω), by Lemma 2.2, there exists 𝐅∈𝐻(div,Ω) such that ∇⋅𝐅=𝑢_𝑡𝑡−𝑓. Thus, (2.19) reduces to (∇⋅𝐩,∇⋅𝐪)=(∇⋅𝐅,∇⋅𝐪),∀𝐪∈𝐻(div,Ω).(2.20) Recalling Lemma 2.1, one concludes that ∇⋅𝐅=∇⋅𝐩,(2.21) that is, 𝑢_𝑡𝑡−∇⋅𝐩=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 𝑇_ℎ be a quasi-uniform family of subdivision of domain Ω; that is, Ω=∪_{𝐾∈𝑇ℎ}𝐾 with ℎ = max {diam(𝐾)∶𝐾∈𝑇_ℎ}, and let 𝑉_ℎ be the finite-dimensional subspaces of 𝐻¹⁰(Ω) defined by 𝑉_ℎ=𝑣_ℎ∈𝐻¹⁰(Ω);𝑣_ℎ∣_𝐾∈𝑃_𝑚(𝐾),(2.23) where 𝑃_𝑚(𝐾) denotes the space of polynomials of degree at most 𝑚 on 𝐾. Moreover, we denote the vector space in mixed finite element spaces with index 𝑘 by 𝐻_ℎ. It is well known that both 𝐻_ℎ and 𝑉_ℎ satisfy the inverse property and the following approximation properties [26, 27]: inf_{𝑣ℎ∈𝑉ℎ}‖‖i𝑣nf−_{𝐪ℎ∈𝑊ℎ}𝑣_ℎ‖‖+ℎ‖‖𝑣−𝑣_ℎ‖‖₁≤𝐶ℎ^𝑚+1‖𝑣‖_𝑚+1,𝑣∈𝐻^𝑚+1(Ω),‖‖𝐪−𝐪_ℎ‖‖≤𝐶ℎ^𝑘+1‖𝐪‖_𝑘+1,𝐪_ℎ∈𝐻^𝑘+1(Ω)^𝑑.(2.24)

Let Π_ℎ∶𝐻→𝐻_ℎ denote the Raviart-Thomas interpolation operator [28] which satisfies ∇⋅𝐪−Π_ℎ𝐪,∇⋅𝐪_ℎ=0,∀𝐪_ℎ∈𝐻_ℎ,(2.25) and the following estimates [26, 28, 29] ‖‖𝐪−Π_ℎ𝐪‖‖≤𝐶ℎ^𝑘+1‖𝐪‖_𝑘+1,(2.26)‖‖∇⋅𝐪−Π_ℎ𝐪‖‖≤𝐶ℎ^𝑘‖𝐪‖_𝑘+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 (𝑢_ℎ,𝝈_ℎ,𝐩_ℎ)∈𝑉_ℎ×𝐻_ℎ×𝐻_ℎ such that 𝝈_ℎ𝑡𝑡,𝐪_ℎ+𝝈_ℎ∇⋅𝐩_ℎ,∇⋅𝐪_ℎ=−𝑓,∇⋅𝐪_ℎ,∀𝐪_ℎ∈𝐻_ℎ,𝐩_ℎ,∇𝑣_ℎ=∇𝑢_ℎ,∇𝑣_ℎ,∀𝑣_ℎ∈𝑉_ℎ,,𝐰_ℎ=𝑎𝑢_ℎ𝐩𝝈_ℎ𝑡ℎ,𝐰_ℎ+𝑏𝑢_ℎ𝝈_ℎ,𝐰_ℎ,∀𝐰_ℎ∈𝐻_ℎ,𝝈(𝑥,0)=Π_ℎ𝐩(𝑥,0),_ℎ𝝈(𝑥,0)=Π_ℎ∇𝑢₀(𝑥),_ℎ𝑡(𝑥,0)=Π_ℎ∇𝑢₁(𝑥).(2.28)

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

In fact, set 𝑉_ℎ=span{𝜑_𝑖}^𝑁𝑖=1 and 𝐻_ℎ=span{𝜓_𝑗}^𝑀𝑗=1. Then 𝝈_ℎ,𝐩_ℎ∈𝐻_ℎ and 𝑢_ℎ∈𝑉_ℎ, and; hence, 𝝈_ℎ=^𝑀_𝑗=1𝑝_𝑖(𝑡)𝜓_𝑖(𝑥),𝐩_ℎ=^𝑀_𝑗=1𝜆_𝑖(𝑡)𝜓_𝑖(𝑥),𝑢_ℎ=^𝑁_𝑖=1𝑢_𝑖(𝑡)𝜑_𝑖(𝑥).(2.29) Taking 𝐪_ℎ=𝜓_𝑗, 𝐰_ℎ=𝜓_𝑗, 𝑗=1,2,…,𝑀, 𝑣_ℎ=𝜑_𝑖, 𝑖=1,2,…,𝑁 in (2.28) leads to 𝐴Λ=𝐴𝑀𝑃𝐷_𝑡𝑡(𝑈𝑈+)=𝐵𝑃Λ_𝑡𝐶𝑃=,𝐹,+𝑁(𝑈)𝑃,(2.30) where𝐵=𝐴∇=⋅𝜓𝜓_𝑖𝑖(𝑥),𝜓_𝑗(𝑥)_𝑀×𝑀,𝑃=𝑝₁,𝑝₂,…,𝑝_𝑀^𝑇,𝐷=∇𝜑_𝑖(𝑥),∇⋅𝜓_𝑗(𝑥)_𝑀×𝑀,Λ=𝜆₁,𝜆₂,…,𝜆_𝑀^𝑇,𝐶=𝜓_𝑖(𝑥),∇𝜑_𝑗(𝑥)_𝑁×𝑁,𝑈=𝑢₁,𝑢₂,…,𝑢_𝑁^𝑇,𝑁(𝑈)=(𝑥)𝑏,∇(𝑈𝜑)_𝑗𝜓(𝑥_𝑖)_𝑁×𝑀,𝑀(𝑈)=𝑎(𝑈)𝜓_𝑖(𝑥),𝜓_𝑗(𝑥)_𝑀×𝑀,(𝑥),𝜓_𝑗(𝑥)_𝑀×𝑀,𝐹=−𝑓,∇⋅𝜓_𝑗(𝑥)_𝑀×1,(2.31) and 𝑃(0), 𝑃_𝑡(0) are given.

Note that matrix 𝐴 in (2.31) is positive definite. Thus, by the third equation in (2.30), one has Λ=𝐴⁻¹𝑀𝑃_𝑡+𝑁𝑃.(2.32) Inserting the above equality into the first equation of (2.30) yields 𝐴𝑃_𝑡𝑡+𝐵𝐴⁻¹𝑀𝑃_𝑡+𝐵𝐴⁻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 𝑈 and Λ 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 𝑅_ℎ∶𝐻¹⁰(Ω)→𝑉_ℎ be the Ritz projection defined by ∇𝑢−𝑅_ℎ𝑢,∇𝑣_ℎ=0,∀𝑣_ℎ∈𝑉_ℎ.(3.1) Then the following approximation holds [27]: ‖‖𝑢−𝑅_ℎ𝑢‖‖+ℎ‖‖∇𝑢−𝑅_ℎ𝑢‖‖≤𝐶ℎ^𝑚+1‖𝑢‖_𝑚+1.(3.2)

Let 𝝈𝐩−−𝝈𝐩_ℎℎ=𝐩−Π_ℎ𝐩+Π_ℎ𝐩−𝐩_ℎ=𝜂+𝜁,𝑢−𝑢_ℎ=𝝈−Π_ℎ𝝈+Π_ℎ𝝈−𝝈_ℎ=𝜃+𝜉,=𝑢−𝑅_ℎ𝑢+𝑅_ℎ𝑢−𝑢_ℎ=𝛼+𝛽.(3.3) Utilizing (2.3), (2.28), and auxiliary projections (3.1), (2.25), we can obtain the following error equations:𝜉_𝑡𝑡,𝐪_ℎ+∇⋅𝜁,∇⋅𝐪_ℎ=−𝜃_𝑡,𝐪_ℎ,∀𝐪_ℎ∈𝐻_ℎ,(3.4)𝜉,∇𝑣_ℎ=∇𝛽,∇𝑣_ℎ−𝜃,∇𝑣_ℎ,∀𝑣_ℎ∈𝑉_ℎ,(3.5)𝜁,𝐰_ℎ−𝑎𝑢_ℎ𝜉_𝑡,𝐰_ℎ−𝑏𝑢_ℎ𝜉,𝐰_ℎ=+𝝈_𝑡𝑎𝑎𝑢(_ℎ𝑢)−𝑎𝑢_ℎ,𝐰_ℎ+𝝈𝑏(𝑢)−𝑏𝑢_ℎ,𝐰_ℎ𝜃_𝑡,𝐰_ℎ+𝑏𝑢_ℎ𝜃,𝐰_ℎ−𝜂,𝐰_ℎ,∀𝐰_ℎ∈𝐻_ℎ.(3.6)

Theorem 3.1. Let (𝑢,𝝈,𝐩) and (𝑢_ℎ,𝝈_ℎ,𝐩_ℎ) be the solutions to (2.3) and (2.28), respectively. Then the following error estimates hold: ((𝑎𝑏))‖‖‖‖𝑢∇−⋅𝑢_ℎ𝝈‖‖−₁𝝈≤_ℎ𝐶ℎ^{min(𝑘+1,𝑚)},(𝑐)‖‖𝑢−𝑢_ℎ‖‖≤𝐶ℎ^{min(𝑘,𝑚+1)},‖‖+‖‖𝝈−𝝈_ℎ‖‖+‖‖𝐩−𝐩_ℎ‖‖≤𝐶ℎ^{min(𝑘+1,𝑚+1)},(3.7) where 𝑘≥1 and 𝑚≥1 for 𝑑=2,3, and the positive constant 𝐶 depends on ‖𝑢_𝑡‖_{𝐿∞(𝐻𝑚+1)}, ‖𝑢‖_𝐿^∞_(𝐻^𝑚+1₎, ‖𝐩_𝑡‖_{𝐿∞(𝐻𝑘+1)}, ‖𝐩‖_𝐿^∞_(𝐻^𝑘+1₎, ‖𝝈_𝑡‖_{𝐿∞(𝐻𝑘+1)}, ‖𝝈_𝑡𝑡‖_{𝐿∞(𝐻𝑘+1)}, ‖𝝈‖_𝐿^∞_(𝐻^𝑘+1₎.

Proof. Since estimates of 𝜃, 𝜂, and 𝛼 can be obtained by (3.2) and (2.26), it suffices to estimate 𝜉, 𝜁, and 𝛽.
Instituting 𝐰_ℎ=𝜉_𝑡𝑡 into (3.6) and 𝐪_ℎ=𝜁 in (3.4) gives (∇⋅𝜁,∇⋅𝜁)+𝑎𝑢_ℎ𝜉_𝑡,𝜉_𝑡𝑡+𝑏𝑢_ℎ𝜉,𝜉_𝑡𝑡=−−𝑎𝝈_𝑡𝑢_ℎ𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝑡−𝝈𝑏(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡𝑡𝜃_𝑡,𝜉_𝑡𝑡−𝑏𝑢_ℎ𝜃,𝜉_𝑡𝑡+𝜂,𝜉_𝑡𝑡−𝜃_𝑡𝑡,𝜁.(3.8) It is easy to check that 𝑎𝑏𝑢𝑢_ℎℎ𝜉_𝑡𝑡,𝜉_𝑡=12𝑑𝑑𝑡𝑎𝑢^ℎ𝜉_𝑡,𝜉_𝑡−12𝑎^𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡,𝜂𝜉,,𝜉𝜉_{𝑡𝑡𝑡𝑡}=𝑑𝑑𝑡𝑏𝑢^ℎ𝜉,𝜉_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡−𝑏𝑢_ℎ𝜉_𝑡,𝜉_𝑡,𝝈_𝑡=𝑑𝑑𝑡𝜂,𝜉^𝑡−𝜂_𝑡,𝜉_𝑡,𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝑡=−𝑑𝑑𝑡𝝈𝝈^𝑡_𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡−𝝈_𝑡𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝝈𝑏(𝑢)−𝑏𝑢_ℎ𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡,,𝜉_𝑡𝑡=−𝑑𝑑𝑡𝝈𝝈𝑏_𝑢𝑏(𝑢)−𝑏𝑢^ℎ,𝜉_𝑡−𝝈_𝑡𝑏(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡𝑎𝑢_ℎ(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡,𝑏𝑢_ℎ𝜃_𝑡,𝜉_𝑡𝑡=𝑑𝑑𝑡𝑎𝑢^ℎ𝜃_𝑡,𝜉_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡−𝑎𝑢_ℎ𝜃_𝑡𝑡,𝜉_𝑡,𝜃,𝜉_𝑡𝑡=𝑑𝑑𝑡𝑏𝑢^ℎ𝜃,𝜉_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡−𝑏𝑢_ℎ𝜃_𝑡,𝜉_𝑡.(3.9) Thus, (3.8) can be written as (∇⋅𝜁,∇⋅𝜁)+12𝑑𝑑𝑡𝑎𝑢^ℎ𝜉_𝑡,𝜉_𝑡+𝑑=1𝑑𝑡𝑏𝑢^ℎ𝜉,𝜉_𝑡2𝑎^𝑢𝑢_ℎ𝑢_ℎ𝑡+𝑑𝜉_𝑡,𝜉_𝑡+𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡+𝑏𝑢_ℎ𝜉_𝑡,𝜉_𝑡−𝜃_𝑡𝑡,𝜁𝑑𝑡𝜂,𝜉^𝑡−𝜂_𝑡,𝜉_𝑡−𝑑+𝝈_𝑡𝑡𝑑𝑡𝝈^𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡−𝑑𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡+𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡+𝑑𝝈𝑡𝝈𝑏_𝑢𝑏(𝑢)−𝑏𝑢^ℎ,𝜉_𝑡+𝝈_𝑡𝑏(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡−𝑑𝑑𝑡𝑎𝑢^ℎ𝜃_𝑡+𝑎𝑢_ℎ,𝜉_𝑡+𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡𝜃_𝑡𝑡,𝜉_𝑡−𝑑𝑑𝑡𝑏𝑢^ℎ𝜃,𝜉_𝑡+𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡+𝑏𝑢_ℎ𝜃_𝑡,𝜉_𝑡.(3.10) Integrating this system from 0 to 𝑡 yields ^𝑡0‖∇⋅𝜁‖²𝑑𝜏+12𝑎𝑢^ℎ𝜉_𝑡=𝜂,𝜉_𝑡,𝜉_𝑡+𝑏𝑢_ℎ𝜉,𝜉_𝑡−𝑏𝑢_ℎ−𝝈_𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡−𝝈𝑏(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡−𝑎𝑢_ℎ𝜃_𝑡,𝜉_𝑡−𝜃,𝜉_𝑡+12^𝑡0𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡𝑑𝜏+^𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡𝑑𝜏+^𝑡0𝑏𝑢_ℎ𝜉_𝑡,𝜉_𝑡𝑑𝜏_𝑡0+𝜃_𝑡𝑡,𝜁𝑑𝜏−^𝑡0𝜂_𝑡,𝜉_𝑡𝑑𝜏+^𝑡0𝝈_𝑡𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝑑𝜏_𝑡0+𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑑𝜏+^𝑡0𝝈_𝑡𝑏(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡𝑑𝜏_𝑡0+𝝈𝑏_𝑢(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑑𝜏+^𝑡0𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡𝑑𝜏_𝑡0𝑎𝑢_ℎ𝜃_𝑡𝑡,𝜉_𝑡𝑑𝜏+^𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡𝑑𝜏+^𝑡0𝑏𝑢_ℎ𝜃_𝑡,𝜉_𝑡𝑑𝜏.(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: ||||^𝑡012𝑎^𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡𝑑𝜏||||=12||||^𝑡0𝑎_𝑢𝑢_ℎ𝑢_𝑡𝜉_𝑡,𝜉_𝑡𝑑𝜏+^𝑡012𝑎^𝑢𝑢_ℎ𝑢_ℎ𝑡≤𝐶_𝑡0−𝑢_𝑡𝜉_𝑡,𝜉_𝑡𝑑𝜏||||‖‖𝜉_𝑡‖‖²𝑑𝜏+𝐶‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏.(3.12) For the seventh term on the right-hand side of (3.11), one has ||||^𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡𝑑𝜏||||=≤𝐶||||^𝑡0_𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡−𝑢_𝑡𝜉,𝜉_𝑡+𝑏_𝑢𝑢_ℎ𝑢_𝑡𝜉,𝜉_𝑡𝑑𝜏||||‖‖𝜉^𝑡‖‖²+‖𝜉‖²𝑑𝜏+𝐶‖‖𝜉^𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖𝜉‖²𝑑𝜏.(3.13) For the term ∫^𝑡0(𝝈_𝑡(𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢(𝑢_ℎ)𝑢_ℎ𝑡),𝜉_𝑡)𝑑𝜏 on the right side of (3.11), we have ||||^𝑡0𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑑𝜏||||=≤𝐶||||^𝑡0_𝑡0𝝈_𝑡𝑎_𝑢(𝑢)−𝑎_𝑢𝑢_ℎ𝑢_𝑡+𝑎_𝑢𝑢_ℎ𝑢_𝑡−𝑢_ℎ𝑡,𝜉_𝑡𝑑𝜏||||‖𝛼‖²+‖𝛽‖²+‖‖𝛼_𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏.(3.14) Similarly, ||||^𝑡0𝝈𝑏_𝑢(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑑𝜏||||≤≤𝐶^𝑡0||_𝑡0𝝈𝑏_𝑢(𝑢)−𝑏_𝑢𝑢_ℎ𝑢_𝑡+𝑏_𝑢𝑢_ℎ𝑢_𝑡−𝑢_ℎ𝑡,𝜉_𝑡||𝑑𝜏||||‖𝛼‖²+‖𝛽‖²+‖‖𝛼_𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏,_𝑡0𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡𝑑𝜏||||=≤𝐶||||^𝑡0‖‖𝜉𝑎_𝑢𝑡𝑢_ℎ𝑢_ℎ𝑡−𝑢_𝑡𝜃_𝑡,𝜉_𝑡𝑑𝜏+^𝑡0𝑎_𝑢𝑢_ℎ𝑢_𝑡𝜃_𝑡,𝜉_𝑡𝑑𝜏||||‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡+𝐶_𝑡0‖‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜃_𝑡‖‖²𝑑𝜏||||‖‖𝜃^𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏,_𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡𝑑𝜏||||≤≤𝐶^𝑡0‖‖||𝜉_𝑡𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡−𝑢_𝑡𝜃,𝜉_𝑡||𝑑𝜏+^𝑡0||𝑏_𝑢𝑢_ℎ𝑢_𝑡𝜃,𝜉_𝑡||𝑑𝜏‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡+𝐶_𝑡0‖‖²+‖‖𝛽_𝑡‖‖²+‖𝜃‖²𝑑𝜏‖𝜃‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏.(3.15)
Inserting (3.12)–(3.15) into (3.11) and using the Cauchy-Schwartz inequality lead to ^𝑡0‖∇⋅𝜁‖²𝑑𝜏+12𝑎𝑢^ℎ𝜉_𝑡≤𝐶‖𝜂‖₂,𝜉_𝑡+𝑏𝑢_ℎ𝜉,𝜉_𝑡+𝐶_𝑡0+‖‖𝜉_𝑡‖‖²+‖𝛼‖²+‖𝛽‖²+‖𝜃‖²+‖‖𝜃_𝑡‖‖²+𝐶‖‖𝜉_𝑡‖‖𝜃^𝑡𝑡‖‖²+‖‖𝜃_𝑡‖‖²+‖‖𝜉_𝑡‖‖²+‖‖𝜂_𝑡‖‖²+‖𝛼‖²+‖𝛽‖²+‖𝜁‖²𝑑𝜏‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡+𝐶‖‖𝜉_𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝜃^𝑡+𝐶_𝑡0‖‖²+‖𝜃‖²+‖𝜉‖²𝑑𝜏‖‖𝜃^𝑡‖‖²+‖𝜃‖²+‖𝜉‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏.(3.16) Integrating (3.16) from 0 to 𝑡, using the fact (𝑏(𝑢_ℎ)𝜉,𝜉_𝑡)=(1/2)(𝑑/𝑑𝑡)(𝑏(𝑢_ℎ)𝜉,𝜉)−(1/2)(𝑏_𝑢(𝑢_ℎ)𝑢_ℎ𝑡𝜉,𝜉_𝑡) and the inequality ^𝑡0^𝜏0||𝜓(𝑠)||²𝑑𝑠𝑑𝜏≤𝐶^𝑡0||𝜓(𝑠)||²𝑑𝑠,(3.17) yields ‖𝜉‖²≤𝐶‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡+𝐶‖‖𝜉_𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝜃^𝑡+𝐶_𝑡0‖‖²+‖𝜃‖²+‖𝜉‖²𝑑𝜏‖‖𝛼^𝑡‖‖²+‖‖𝛽_𝑡‖‖²+‖𝛼‖²+‖𝛽‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖‖𝜃_𝑡𝑡‖‖²+‖‖𝜂_𝑡‖‖²+‖𝜉‖²+‖‖𝜉_𝑡‖‖²+‖𝜁‖²𝑑𝜏.(3.18) Thus, to estimate ‖𝜉‖, we need to estimate ‖𝛽‖, ‖𝛽_𝑡‖, ‖𝜁‖, and ‖𝜉_𝑡‖. Taking 𝑣_ℎ=𝛽 in (3.5) leads to (∇𝛽,∇𝛽)=(𝜉,∇𝛽)+(𝜃,∇𝛽).(3.19) By the Cauchy-Schwartz inequality, we obtain ‖∇𝛽‖≤𝐶(‖𝜉‖+‖𝜃‖).(3.20) Note that 𝛽∈𝑉_ℎ⊂𝐻¹⁰(Ω) and ‖𝛽‖≤𝐶‖∇𝛽‖. We further have ‖𝛽‖≤𝐶(‖𝜉‖+‖𝜃‖).(3.21) Differentiating (3.5) with respect to 𝑡 and choosing 𝑣_ℎ=𝛽_𝑡 gives ‖‖∇𝛽_𝑡‖‖≤𝐶‖‖𝜉_𝑡‖‖+‖‖𝜃_𝑡‖‖.(3.22) Similarly, since 𝛽∈𝑉_ℎ⊂𝐻¹⁰(Ω), one has ‖𝛽_𝑡‖≤‖∇𝛽_𝑡‖≤𝐶(‖𝜉_𝑡‖+‖𝜃_𝑡‖).
Taking 𝐰_ℎ=𝜁 in (3.6), one has (𝜁,𝜁)=+𝑎𝝈𝑢_ℎ𝑏(𝜉𝑢_𝑡),𝜁−𝑏+𝑢_ℎ𝑏𝑢_ℎ𝜉,𝜁+𝝈_𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜁,𝜁+𝑎𝑢_ℎ𝜃_𝑡,𝜁+𝑏𝑢_ℎ𝜃,𝜁−(𝜂,𝜁).(3.23) By the Cauchy-Schwartz inequality, we obtain ‖𝜁‖≤𝐶‖𝜉‖+‖‖𝜃_𝑡‖‖+‖𝜃‖+‖𝜂‖+‖𝛼‖+‖𝛽‖+‖‖𝜉_𝑡‖‖.(3.24) To bound ‖𝜉_𝑡‖², we differentiate (3.6) with respect to 𝑡 to obtain 𝜁_𝑡,𝐰=_ℎ𝑎−_𝑢𝑎𝑢_ℎ𝜉_𝑡𝑡,𝐰_ℎ−𝑏𝑢_ℎ𝜉_𝑡,𝐰_ℎ+𝝈𝑢_ℎ𝑡𝑡𝑢_ℎ𝑡𝜉_𝑡,𝐰_ℎ+𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝐰_ℎ+𝝈_𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝐰_ℎ+𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝐰_ℎ+𝑎𝑢𝑏_ℎ(𝑢)−𝑏𝑢_ℎ,𝐰_ℎ+𝝈𝑏_𝑢(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝐰_ℎ+𝑏𝑢_ℎ𝜃_𝑡𝑡,𝐰_ℎ+𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝐰_ℎ𝜃_𝑡,𝐰_ℎ+𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝐰_ℎ−𝜂_𝑡,𝐰_ℎ,∀𝐰_ℎ∈𝐻_ℎ.(3.25) Testing (3.25) with 𝐰_ℎ=𝜉_𝑡𝑡 and (3.4) with 𝐪_ℎ=𝜁_𝑡 and combining the resulting equations together lead to ∇⋅=𝜁,−∇⋅𝑎𝜁_𝑡𝑢+𝑎𝑢_ℎ𝜉_𝑡𝑡,𝜉_𝑡𝑡+𝑏𝑢_ℎ𝜉_𝑡,𝜉_𝑡𝑡−𝝈_𝑡𝑡𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡𝑡−𝝈_𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝑡−𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑡−𝑎𝑢𝑏_ℎ(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡𝑡−𝝈𝑏_𝑢(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑡−𝑏𝑢_ℎ𝜃_𝑡𝑡,𝜉_𝑡𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡𝑡−𝜃_𝑡,𝜁_𝑡𝜃_𝑡,𝜉_𝑡𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡𝑡+𝜂_𝑡,𝜉_𝑡𝑡.(3.26) Note that 𝑏𝑢_ℎ𝜉_𝑡,𝜉_𝑡𝑡∇=⋅𝜁12,∇𝑑𝑑𝑡⋅𝜁𝑏_𝑡𝑢^ℎ𝜉_𝑡,𝜉_𝑡−12𝑏^𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡,𝜃=12𝑑𝑑𝑡(∇⋅𝜁,∇⋅𝜁),_𝑡,𝜁_𝑡=𝑑𝑑𝑡𝜃^𝑡,𝜁−𝜃_𝑡𝑡,𝜁.(3.27) Thus, (3.26) can be rewritten as 12𝑑𝑑𝑡(∇⋅𝜁,∇⋅𝜁)+𝑎𝑢^ℎ𝜉_𝑡𝑡,𝜉_𝑡𝑡+12𝑑𝑑𝑡𝑏𝑢^ℎ𝜉_𝑡=−𝑎_𝑢,𝜉_𝑡−𝝈_𝑡𝑡𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡𝑡−12𝑏^𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡𝑡−𝝈_𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝑡−𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑡−𝑎𝑏𝑢(𝑢_ℎ)−𝑏𝑢_ℎ,𝜉_𝑡𝑡−𝝈𝑏_𝑢(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑡−𝑏𝑢𝜃_𝑡𝑡,𝜉_𝑡𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡𝑡−𝑑𝑑𝑡𝜃^𝑡,𝜁+𝜃_𝑡𝑡,𝜁_ℎ𝜃_𝑡,𝜉_𝑡𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡𝑡+𝜂_𝑡,𝜉_𝑡𝑡.(3.28) Integrating (3.28) from 0 to 𝑡 yields (∇⋅=𝜁,−∇⋅_𝑡0𝜁)+^𝑡0𝑎𝑢_ℎ𝜉_𝑡𝑡,𝜉_𝑡𝑡+𝑏𝑢_ℎ𝜉_𝑡,𝜉_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡𝑡𝑑𝜏−12^𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉,𝜉_𝑡𝑡𝑑𝜏_𝑡0−𝝈_𝑡𝑡𝑎(𝑢)−𝑎𝑢_ℎ,𝜉_𝑡𝑡𝑑𝜏−^𝑡0𝝈_𝑡𝑎_𝑢(𝑢)𝑢_𝑡−𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑡𝑑𝜏_𝑡0−𝝈_𝑡𝑏(𝑢)−𝑏𝑢_ℎ,𝜉_𝑡𝑡𝑑𝜏−^𝑡0𝝈𝑏_𝑢(𝑢)𝑢_𝑡−𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡,𝜉_𝑡𝑡𝑑𝜏_𝑡0−𝑎𝑢_ℎ𝜃_𝑡𝑡,𝜉_𝑡𝑡𝑑𝜏−^𝑡0𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃_𝑡,𝜉_𝑡𝑡𝑑𝜏−𝜃_𝑡,𝜁+^𝑡0𝜃_𝑡𝑡,𝜁𝑑𝜏_𝑡0𝑏𝑢_ℎ𝜃_𝑡,𝜉_𝑡𝑡𝑑𝜏−^𝑡0𝑏_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜃,𝜉_𝑡𝑡𝑑𝜏+^𝑡0𝜂_𝑡,𝜉_𝑡𝑡𝑑𝜏.(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 𝜀>0, it holds that ||||−^𝑡0𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡𝜉_𝑡,𝜉_𝑡𝑡𝑑𝜏||||≤≤𝐶||||‖‖^𝑡0𝜉_𝑡𝑎_𝑢𝑢_ℎ𝑢_ℎ𝑡−𝑢_𝑡𝜉_𝑡,𝜉_𝑡𝑡𝑑𝜏||||+||||^𝑡0𝑎_𝑢𝑢_ℎ𝑢_𝑡𝜉_𝑡,𝜉_𝑡𝑡𝑑𝜏||||‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡+𝐶_𝑡0‖‖²+‖‖𝛽_𝑡‖‖²𝑑𝜏‖‖𝜉_𝑡‖‖²𝑑𝜏+𝜀1+‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝜉_𝑡𝑡‖‖²𝑑𝜏.(3.30) Similarly, we can bound (3.29) as follows: ‖∇⋅𝜁‖²+‖‖𝜉_𝑡‖‖²+^𝑡0‖‖𝜉_𝑡𝑡‖‖²𝑑𝜏≤𝐶‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝛼^𝑡+𝜀1+‖‖𝜉_𝑡‖‖²+‖‖𝛽_𝑡‖‖²𝑑𝜏+𝐶‖𝜉‖‖‖_{𝐿∞(0,𝑡;𝐿∞)𝐿∞(0,𝑡;𝐿∞)}^𝑡0‖‖𝜉_𝑡𝑡‖‖²𝑑𝜏^𝑡0‖‖𝛼^𝑡+𝜀1+‖𝜉‖_{𝐿∞(0,𝑡;𝐿∞)}‖‖²+‖‖𝛽_𝑡‖‖²𝑑𝜏+𝐶_𝑡0^𝑡0‖‖𝜉_𝑡𝑡‖‖²𝑑𝜏‖𝜉‖²+‖𝜁‖²+‖‖𝜃_𝑡𝑡‖‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖‖𝜃+‖𝛼‖²+‖‖𝛼_𝑡‖‖²+‖𝛽‖²+‖‖𝛽_𝑡‖‖²+‖‖𝜂_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏_𝑡‖‖‖𝜁‖.(3.31) In the following error analysis, we make an induction hypothesis: ‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}+‖𝜉‖_𝐿^∞_(0,𝑡;𝐿^∞₎≤1.(3.32) Utilizing (3.32), (3.24), (3.22), (3.21), and Young's inequality, one can reduce (3.31) to ‖∇⋅𝜁‖²+‖‖𝜉_𝑡‖‖²≤+𝐶𝐶‖_𝑡0𝜉‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖𝜂‖²‖𝜉‖²+‖‖𝜃_𝑡𝑡‖‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖‖𝛼_t‖‖²+‖‖𝜂_𝑡‖‖²+‖‖𝜉_𝑡‖‖²𝑑𝜏.(3.33) Then by Gronwall's inequality, we obtain ‖∇⋅𝜁‖²+‖‖𝜉_𝑡‖‖²≤+𝐶𝐶‖_𝑡0𝜉‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖𝜂‖²‖𝜉‖²+‖‖𝜃_𝑡𝑡‖‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖‖𝛼_𝑡‖‖²+‖‖𝜂_𝑡‖‖²𝑑𝜏.(3.34) Furthermore, by (3.24) and (3.34), one has ‖𝜁‖²≤+𝐶𝐶‖_𝑡0𝜉‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖𝜂‖²‖𝜉‖²+‖‖𝜃_𝑡𝑡‖‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖‖𝛼_𝑡‖‖²+‖‖𝜂_𝑡‖‖²𝑑𝜏.(3.35) Therefore, by the estimates of ‖𝛽‖, ‖𝛽_𝑡‖, ‖𝜁‖, and ‖𝜉_𝑡‖, it follows that ‖𝜉‖²≤𝐶^𝑡0‖𝜉‖²+‖‖𝜃_𝑡𝑡‖‖²+‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖‖𝛼_𝑡‖‖²+‖𝜂‖²+‖‖𝜂_𝑡‖‖²𝑑𝜏.(3.36) Applying Gronwall's inequality to the above equation and using the estimates of projection operators give ‖𝜉‖²≤≤𝐶𝐶ℎ^𝑡0‖‖𝜃^𝑡𝑡‖‖²+_{min(2𝑘+2,2𝑚+2)}‖‖𝜃_𝑡‖‖²+‖𝜃‖²+‖𝛼‖²+‖‖𝛼_𝑡‖‖²+‖𝜂‖²+‖‖𝜂_𝑡‖‖²𝑑𝜏‖‖𝑢^𝑡‖‖^2𝐿_{∞𝐻𝑚+1}+‖𝑢‖^2𝐿_{∞𝐻𝑚+1}+‖‖𝐩_𝑡‖‖^2𝐿_{∞𝐻𝑘+1}+‖𝐩‖^2𝐿_{∞(𝐻𝑘+1)}+‖‖𝝈_𝑡‖‖^2𝐿_{∞(𝐻𝑘+1)}+‖‖𝝈_𝑡𝑡‖‖^2𝐿_{∞(𝐻𝑘+1)}+‖𝝈‖^2𝐿_{∞𝐻𝑘+1}.(3.37) Inserting the estimate of ‖𝜉‖ into (3.34) yields ‖‖𝜉_𝑡‖‖²≤𝐶ℎ^{min(2𝑘+2,2𝑚+2)}.(3.38) Thus, the estimates of 𝛽 and 𝜁 follow from the estimate of 𝜉.
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 𝑡=0, then 𝜉(0)=0, 𝜉_𝑡(0)=0. Note that ‖𝜉‖_𝐿^∞_(0,𝑡;𝐿^∞₎+‖𝜉_𝑡‖_𝐿^∞_(0,𝑡;𝐿^∞₎ is continuous w.r.t. 𝑡. Then, we conclude that there exists 𝑡₁∈(0,𝑇] such that ‖𝜉‖_𝐿^∞_{(0,𝑡1;𝐿}^∞₎+‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡1;𝐿∞)}≤1.(3.39) Set 𝑡^∗=sup𝑡₁. Thus, ‖𝜉‖_𝐿^∞_{(0,𝑡∗;𝐿}^∞₎+‖𝜉_𝑡‖_𝐿^∞_{(0,𝑡∗;𝐿}^∞₎≤1. Therefore, we have ‖‖𝜉𝑡^∗‖‖+‖‖𝜉_𝑡𝑡^∗‖‖≤𝐶ℎ^{min(𝑘+1,𝑚+1)}.(3.40)
By inverse estimates, we deduce that, for any 0≤𝑡≤𝑡^∗, it holds that ‖𝜉‖_𝐿^∞_(0,𝑡;𝐿^∞₎+‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡;𝐿∞)}≤𝐶ℎ^{min(𝑘+1,𝑚+1)−𝑑/2}.(3.41) Then we can take ℎ>0 sufficiently small such that ‖𝜉‖_𝐿^∞_{(0,𝑡∗;𝐿}^∞₎+‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡∗;𝐿∞)}<1.(3.42) Again, by the continuity of ‖𝜉‖_𝐿^∞_(0,𝑡;𝐿^∞₎+‖𝜉_𝑡‖_𝐿^∞_(0,𝑡;𝐿^∞₎, we conclude that there exists a positive constant 𝛿 such that ‖𝜉‖_𝐿^∞_{(0,𝑡∗+𝛿;𝐿}^∞₎+‖‖𝜉_𝑡‖‖_{𝐿∞(0,𝑡∗+𝛿;𝐿∞)}≤1,(3.43) which contracts to the definition of 𝑡^∗. 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: ‖‖𝑢−𝑢_ℎ‖‖_{𝐿∞(𝐿∞)}≤𝐶₂||lnℎ||^𝑑−1ℎ^{min(𝑘+1,𝑚+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).