# An Optimal Error Estimates ofH1-Galerkin Expanded Mixed Finite Element Methods for Nonlinear Viscoelasticity-Type Equation

Authors: Haitao Che; Yiju Wang; Zhaojie Zhou
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

We investigate aH1-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.


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## Body

## 1. Introduction

Consider the following nonlinear viscoelasticity-type equation:(1.1)utt-∇⋅(a(x,u)∇ut+b(x,u)∇u)=f(x,t),(x,t)∈Ω×J,u(x,t)=0,(x,t)∈∂Ω×J,u(x,0)=u0(x),x∈Ω,ut(x,0)=u1(x),x∈Ω,
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 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 ∇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 Section2, 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 useH 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 H01(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 functionsa(x,u),b(x,u),au(x,u), and bu(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 theH1-Galerkin expanded mixed finite element procedure, we introduce vector
(2.1)p=a(x,u)∇ut+b(x,u)∇u,σ=∇u,
and split (1.1) into a first-order system as follows:
(2.2)utt-∇⋅p=f,σ=∇u,p=a(u)σt+b(u)σ,σ(x,0)=∇u0(x),σt(x,0)=∇u1(x),p(x,0)=a(u0)∇u1(x)+b(u0)∇u0(x).
Then by Green's formula we can further define the following weak formulation of problem (2.2): find (u,σ,p)∈H01(Ω)×H(div,Ω)×H(div,Ω) such that
(2.3)(σtt,q)+(∇⋅p,∇⋅q)=-(f,∇⋅q),∀q∈H(div⁡,Ω),(σ,∇v)=(∇u,∇v),∀v∈H01(Ω),(p,w)=(a(u)σt,w)+(b(u)σ,w),∀w∈H(div⁡,Ω),σ(x,0)=∇u0(x),σt(x,0)=∇u1(x),p(x,0)=a(u0)∇u1(x)+b(u0)∇u0(x).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]).
LetΩ be a bounded domain with a Lipschitz continuous boundary ∂Ω. Then, for any p∈H(div,Ω), there exists ϕ∈H2(Ω)⋂H01(Ω) and divergence free ψ∈H(div,Ω) such that ∇·ψ=0 and p=∇ϕ+ψ.Lemma 2.2 (see [26]).
LetΩ 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 Lemmas2.1 and 2.2, (u,σ,p)∈H01(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, takingw=p-a(u)σt-b(u)σ in the third equation of (2.3) leads to
(2.4)(p-a(u)σt-b(u)σ,p-a(u)σt-b(u)σ)=0,
which implies
(2.5)p=a(u)σt-b(u)σ.
By Lemma2.1, there exist ϕ∈H2(Ω)⋂H01(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
(2.6)(∇ϕ+ψ,∇v)=(∇u,∇v),∀v∈H01(Ω).
By the divergence theorem [1], one has
(2.7)(ψ,∇v)=-(∇⋅ψ,v)=0,∀v∈H01(Ω).
Substituting (2.7) into (2.6) yields
(2.8)(∇ϕ,∇v)=(∇u,∇v),∀v∈H01(Ω),
which means that
(2.9)∇ϕ=∇u,σ=∇u+ψ.
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
(2.10)(utt,∇⋅q)-(ψtt,q)-(∇⋅(a(u)(∇ut+ψt)),∇⋅q)+(∇⋅(b(u)(∇u+ψ)),∇⋅q)=(f,∇⋅q).
Instituting q=ψt into (2.10) and using ∇·ψt=0 lead to
(2.11)0=(ψtt,ψt)=12ddt(ψt,ψt).
Integrating from 0 to t with respect to time results in
(2.12)(ψt(x,t),ψt(x,t))=(ψt(x,0),ψt(x,0)).
Differentiating (2.9) with respect to t, one obtains
(2.13)σt=∇ut+ψt.
By the fifth equation in (2.3), we deduce that
(2.14)ψt(x,0)=0,
which implies
(2.15)ψt(x,t)=0.
Integrating the equation ψt(x,t)=0 with respect to t from 0 to t gives
(2.16)ψ(x,t)=ψ(x,0).
By (2.9) and the forth equation in (2.2), we deduce
(2.17)ψ(x,t)=0,
which leads to
(2.18)σ=∇u.
Therefore, (2.10) can equivalently be transformed into the following equation:
(2.19)(utt,∇⋅q)-(∇⋅(a(u)∇ut+b(u)∇u),∇⋅q)=(f,∇⋅q),∀q∈H(div⁡,Ω).
For f,utt∈L2(Ω), by Lemma 2.2, there exists F∈H(div,Ω) such that ∇·F=utt-f. Thus, (2.19) reduces to
(2.20)(∇⋅p,∇⋅q)=(∇⋅F,∇⋅q),∀q∈H(div⁡,Ω).
Recalling Lemma 2.1, one concludes that
(2.21)∇⋅F=∇⋅p,
that is,
(2.22)utt-∇⋅p=f.
Combining this with (2.5) and (2.18) results in the desired assertion, and this completes the proof.


### 2.2. Numerical Scheme

LetTh be a quasi-uniform family of subdivision of domain Ω; that is, Ω=∪K∈ThK with h = max {diam(K):K∈Th}, and let Vh be the finite-dimensional subspaces of H01(Ω) defined by (2.23)Vh={vh∈H01(Ω);vh∣K∈Pm(K)},
where Pm(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 Hh. It is well known that both Hh and Vh satisfy the inverse property and the following approximation properties [26, 27]: (2.24)inf⁡⁡vh∈Vh‖v-vh‖+h‖v-vh‖1≤Chm+1‖v‖m+1,v∈Hm+1(Ω),inf⁡⁡qh∈Wh‖q-qh‖≤Chk+1‖q‖k+1,qh∈(Hk+1(Ω))d.LetΠh:H→Hh denote the Raviart-Thomas interpolation operator [28] which satisfies
(2.25)(∇⋅(q-Πhq),∇⋅qh)=0,∀qh∈Hh,
and the following estimates [26, 28, 29]
(2.26)‖q-Πhq‖≤Chk+1‖q‖k+1,(2.27)‖∇⋅(q-Πhq)‖≤Chk‖q‖k+1.With the above notations, the semidiscreteH1-Galerkin expanded mixed finite element method for system (2.3) is reduced to find a triple (uh,σh,ph)∈Vh×Hh×Hh such that
(2.28)(σhtt,qh)+(∇⋅ph,∇⋅qh)=-(f,∇⋅qh),∀qh∈Hh,(σh,∇vh)=(∇uh,∇vh),∀vh∈Vh,(ph,wh)=(a(uh)σht,wh)+(b(uh)σh,wh),∀wh∈Hh,ph(x,0)=Πhp(x,0),σh(x,0)=Πh∇u0(x),σht(x,0)=Πh∇u1(x).For theH1-Galerkin expanded mixed finite element scheme (2.28), we claim that there exists a unique solution.In fact, setVh=span{φi}i=1N and Hh=span{ψj}j=1M. Then σh,ph∈Hh and uh∈Vh, and; hence,
(2.29)σh=∑j=1Mpi(t)ψi(x),ph=∑j=1Mλi(t)ψi(x),uh=∑i=1Nui(t)φi(x).
Taking qh=ψj, wh=ψj, j=1,2,…,M, vh=φi, i=1,2,…,N in (2.28) leads to
(2.30)APtt+BΛ=F,DU=CP,AΛ=M(U)Pt+N(U)P,
where(2.31)A=(ψi(x),ψj(x))M×M,P=(p1,p2,…,pM)T,B=(∇⋅ψi(x),∇⋅ψj(x))M×M,Λ=(λ1,λ2,…,λM)T,D=(∇φi(x),∇φj(x))N×N,U=(u1,u2,…,uN)T,C=(ψi(x),∇φj(x))N×M,M(U)=(a(U)ψi(x),ψj(x))M×M,N(U)=(b(U)ψi(x),ψj(x))M×M,F=-(f,∇⋅ψj(x))M×1,
and P(0), Pt(0) are given.Note that matrixA in (2.31) is positive definite. Thus, by the third equation in (2.30), one has
(2.32)Λ=A-1(MPt+N)P.
Inserting the above equality into the first equation of (2.30) yields
(2.33)APtt+BA-1MPt+BA-1NP=F.
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.


## 2.1. Weak Formulation

To define theH1-Galerkin expanded mixed finite element procedure, we introduce vector
(2.1)p=a(x,u)∇ut+b(x,u)∇u,σ=∇u,
and split (1.1) into a first-order system as follows:
(2.2)utt-∇⋅p=f,σ=∇u,p=a(u)σt+b(u)σ,σ(x,0)=∇u0(x),σt(x,0)=∇u1(x),p(x,0)=a(u0)∇u1(x)+b(u0)∇u0(x).
Then by Green's formula we can further define the following weak formulation of problem (2.2): find (u,σ,p)∈H01(Ω)×H(div,Ω)×H(div,Ω) such that
(2.3)(σtt,q)+(∇⋅p,∇⋅q)=-(f,∇⋅q),∀q∈H(div⁡,Ω),(σ,∇v)=(∇u,∇v),∀v∈H01(Ω),(p,w)=(a(u)σt,w)+(b(u)σ,w),∀w∈H(div⁡,Ω),σ(x,0)=∇u0(x),σt(x,0)=∇u1(x),p(x,0)=a(u0)∇u1(x)+b(u0)∇u0(x).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]).
LetΩ be a bounded domain with a Lipschitz continuous boundary ∂Ω. Then, for any p∈H(div,Ω), there exists ϕ∈H2(Ω)⋂H01(Ω) and divergence free ψ∈H(div,Ω) such that ∇·ψ=0 and p=∇ϕ+ψ.Lemma 2.2 (see [26]).
LetΩ 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 Lemmas2.1 and 2.2, (u,σ,p)∈H01(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, takingw=p-a(u)σt-b(u)σ in the third equation of (2.3) leads to
(2.4)(p-a(u)σt-b(u)σ,p-a(u)σt-b(u)σ)=0,
which implies
(2.5)p=a(u)σt-b(u)σ.
By Lemma2.1, there exist ϕ∈H2(Ω)⋂H01(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
(2.6)(∇ϕ+ψ,∇v)=(∇u,∇v),∀v∈H01(Ω).
By the divergence theorem [1], one has
(2.7)(ψ,∇v)=-(∇⋅ψ,v)=0,∀v∈H01(Ω).
Substituting (2.7) into (2.6) yields
(2.8)(∇ϕ,∇v)=(∇u,∇v),∀v∈H01(Ω),
which means that
(2.9)∇ϕ=∇u,σ=∇u+ψ.
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
(2.10)(utt,∇⋅q)-(ψtt,q)-(∇⋅(a(u)(∇ut+ψt)),∇⋅q)+(∇⋅(b(u)(∇u+ψ)),∇⋅q)=(f,∇⋅q).
Instituting q=ψt into (2.10) and using ∇·ψt=0 lead to
(2.11)0=(ψtt,ψt)=12ddt(ψt,ψt).
Integrating from 0 to t with respect to time results in
(2.12)(ψt(x,t),ψt(x,t))=(ψt(x,0),ψt(x,0)).
Differentiating (2.9) with respect to t, one obtains
(2.13)σt=∇ut+ψt.
By the fifth equation in (2.3), we deduce that
(2.14)ψt(x,0)=0,
which implies
(2.15)ψt(x,t)=0.
Integrating the equation ψt(x,t)=0 with respect to t from 0 to t gives
(2.16)ψ(x,t)=ψ(x,0).
By (2.9) and the forth equation in (2.2), we deduce
(2.17)ψ(x,t)=0,
which leads to
(2.18)σ=∇u.
Therefore, (2.10) can equivalently be transformed into the following equation:
(2.19)(utt,∇⋅q)-(∇⋅(a(u)∇ut+b(u)∇u),∇⋅q)=(f,∇⋅q),∀q∈H(div⁡,Ω).
For f,utt∈L2(Ω), by Lemma 2.2, there exists F∈H(div,Ω) such that ∇·F=utt-f. Thus, (2.19) reduces to
(2.20)(∇⋅p,∇⋅q)=(∇⋅F,∇⋅q),∀q∈H(div⁡,Ω).
Recalling Lemma 2.1, one concludes that
(2.21)∇⋅F=∇⋅p,
that is,
(2.22)utt-∇⋅p=f.
Combining this with (2.5) and (2.18) results in the desired assertion, and this completes the proof.


## 2.2. Numerical Scheme

LetTh be a quasi-uniform family of subdivision of domain Ω; that is, Ω=∪K∈ThK with h = max {diam(K):K∈Th}, and let Vh be the finite-dimensional subspaces of H01(Ω) defined by (2.23)Vh={vh∈H01(Ω);vh∣K∈Pm(K)},
where Pm(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 Hh. It is well known that both Hh and Vh satisfy the inverse property and the following approximation properties [26, 27]: (2.24)inf⁡⁡vh∈Vh‖v-vh‖+h‖v-vh‖1≤Chm+1‖v‖m+1,v∈Hm+1(Ω),inf⁡⁡qh∈Wh‖q-qh‖≤Chk+1‖q‖k+1,qh∈(Hk+1(Ω))d.LetΠh:H→Hh denote the Raviart-Thomas interpolation operator [28] which satisfies
(2.25)(∇⋅(q-Πhq),∇⋅qh)=0,∀qh∈Hh,
and the following estimates [26, 28, 29]
(2.26)‖q-Πhq‖≤Chk+1‖q‖k+1,(2.27)‖∇⋅(q-Πhq)‖≤Chk‖q‖k+1.With the above notations, the semidiscreteH1-Galerkin expanded mixed finite element method for system (2.3) is reduced to find a triple (uh,σh,ph)∈Vh×Hh×Hh such that
(2.28)(σhtt,qh)+(∇⋅ph,∇⋅qh)=-(f,∇⋅qh),∀qh∈Hh,(σh,∇vh)=(∇uh,∇vh),∀vh∈Vh,(ph,wh)=(a(uh)σht,wh)+(b(uh)σh,wh),∀wh∈Hh,ph(x,0)=Πhp(x,0),σh(x,0)=Πh∇u0(x),σht(x,0)=Πh∇u1(x).For theH1-Galerkin expanded mixed finite element scheme (2.28), we claim that there exists a unique solution.In fact, setVh=span{φi}i=1N and Hh=span{ψj}j=1M. Then σh,ph∈Hh and uh∈Vh, and; hence,
(2.29)σh=∑j=1Mpi(t)ψi(x),ph=∑j=1Mλi(t)ψi(x),uh=∑i=1Nui(t)φi(x).
Taking qh=ψj, wh=ψj, j=1,2,…,M, vh=φi, i=1,2,…,N in (2.28) leads to
(2.30)APtt+BΛ=F,DU=CP,AΛ=M(U)Pt+N(U)P,
where(2.31)A=(ψi(x),ψj(x))M×M,P=(p1,p2,…,pM)T,B=(∇⋅ψi(x),∇⋅ψj(x))M×M,Λ=(λ1,λ2,…,λM)T,D=(∇φi(x),∇φj(x))N×N,U=(u1,u2,…,uN)T,C=(ψi(x),∇φj(x))N×M,M(U)=(a(U)ψi(x),ψj(x))M×M,N(U)=(b(U)ψi(x),ψj(x))M×M,F=-(f,∇⋅ψj(x))M×1,
and P(0), Pt(0) are given.Note that matrixA in (2.31) is positive definite. Thus, by the third equation in (2.30), one has
(2.32)Λ=A-1(MPt+N)P.
Inserting the above equality into the first equation of (2.30) yields
(2.33)APtt+BA-1MPt+BA-1NP=F.
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 theH1-Galerkin expanded mixed finite element method.For error analysis in the following, we need to introduce a projection operator. LetRh:H01(Ω)→Vh be the Ritz projection defined by
(3.1)(∇(u-Rhu),∇vh)=0,∀vh∈Vh.
Then the following approximation holds [27]:
(3.2)‖u-Rhu‖+h‖∇(u-Rhu)‖≤Chm+1‖u‖m+1.Let(3.3)p-ph=(p-Πhp)+(Πhp-ph)=η+ζ,σ-σh=(σ-Πhσ)+(Πhσ-σh)=θ+ξ,u-uh=(u-Rhu)+(Rhu-uh)=α+β.
Utilizing (2.3), (2.28), and auxiliary projections (3.1), (2.25), we can obtain the following error equations:(3.4)(ξtt,qh)+(∇⋅ζ,∇⋅qh)=-(θt,qh),∀qh∈Hh,(3.5)(ξ,∇vh)=(∇β,∇vh)-(θ,∇vh),∀vh∈Vh,(3.6)(ζ,wh)-(a(uh)ξt,wh)-(b(uh)ξ,wh)=(σt(a(u)-a(uh)),wh)+(σ(b(u)-b(uh)),wh)+(a(uh)θt,wh)+(b(uh)θ,wh)-(η,wh),∀wh∈Hh.Theorem 3.1.
Let(u,σ,p) and (uh,σh,ph) be the solutions to (2.3) and (2.28), respectively. Then the following error estimates hold:
(3.7)(a)‖u-uh‖1≤Chmin⁡⁡(k+1,m),(b)‖∇⋅(σ-σh)‖≤Chmin⁡⁡(k,m+1),(c)‖u-uh‖+‖σ-σh‖+‖p-ph‖≤Chmin⁡⁡(k+1,m+1),
where k≥1 and m≥1 for d=2,3, and the positive constant C depends on ∥ut∥L∞(Hm+1), ∥u∥L∞(Hm+1), ∥pt∥L∞(Hk+1), ∥p∥L∞(Hk+1), ∥σt∥L∞(Hk+1), ∥σtt∥L∞(Hk+1), ∥σ∥L∞(Hk+1).Proof.
Since estimates ofθ, η, and α can be obtained by (3.2) and (2.26), it suffices to estimate ξ, ζ, and β.
Institutingwh=ξtt into (3.6) and qh=ζ in (3.4) gives
(3.8)(∇⋅ζ,∇⋅ζ)+(a(uh)ξt,ξtt)+(b(uh)ξ,ξtt)=-(σt(a(u)-a(uh)),ξtt)-(σ(b(u)-b(uh)),ξtt)-(a(uh)θt,ξtt)-(b(uh)θ,ξtt)+(η,ξtt)-(θtt,ζ).
It is easy to check that
(3.9)(a(uh)ξtt,ξt)=12ddt(a(uh)ξt,ξt)-12(au(uh)uhtξt,ξt),(b(uh)ξ,ξtt)=ddt(b(uh)ξ,ξt)-(bu(uh)uhtξ,ξt)-(b(uh)ξt,ξt),(η,ξtt)=ddt(η,ξt)-(ηt,ξt),(σt(a(u)-a(uh)),ξtt)=ddt(σt(a(u)-a(uh)),ξt)-(σtt(a(u)-a(uh)),ξt)-(σt(au(u)ut-au(uh)uht),ξt),(σ(b(u)-b(uh)),ξtt)=ddt(σ(b(u)-b(uh)),ξt)-(σt(b(u)-b(uh)),ξt)-(σ(bu(u)ut-bu(uh)uht),ξt),(a(uh)θt,ξtt)=ddt(a(uh)θt,ξt)-(au(uh)uhtθt,ξt)-(a(uh)θtt,ξt),(b(uh)θ,ξtt)=ddt(b(uh)θ,ξt)-(bu(uh)uhtθ,ξt)-(b(uh)θt,ξt).
Thus, (3.8) can be written as
(3.10)(∇⋅ζ,∇⋅ζ)+12ddt(a(uh)ξt,ξt)+ddt(b(uh)ξ,ξt)=12(au(uh)uhtξt,ξt)+(bu(uh)uhtξ,ξt)+(b(uh)ξt,ξt)-(θtt,ζ)+ddt(η,ξt)-(ηt,ξt)-ddt(σt(a(u)-a(uh)),ξt)+(σtt(a(u)-a(uh)),ξt)+(σt(au(u)ut-au(uh)uht),ξt)-ddt(σ(b(u)-b(uh)),ξt)+(σt(b(u)-b(uh)),ξt)+(σ(bu(u)ut-bu(uh)uht),ξt)-ddt(a(uh)θt,ξt)+(au(uh)uhtθt,ξt)+(a(uh)θtt,ξt)-ddt(b(uh)θ,ξt)+(bu(uh)uhtθ,ξt)+(b(uh)θt,ξt).
Integrating this system from 0 to t yields
(3.11)∫0t‖∇⋅ζ‖2dτ+12(a(uh)ξt,ξt)+(b(uh)ξ,ξt)=(η,ξt)-(σt(a(u)-a(uh)),ξt)-(σ(b(u)-b(uh)),ξt)-(a(uh)θt,ξt)-(b(uh)θ,ξt)+12∫0t(au(uh)uhtξt,ξt)dτ+∫0t(bu(uh)uhtξ,ξt)dτ+∫0t(b(uh)ξt,ξt)dτ-∫0t(θtt,ζ)dτ-∫0t(ηt,ξt)dτ+∫0t(σtt(a(u)-a(uh)),ξt)dτ+∫0t(σt(au(u)ut-au(uh)uht),ξt)dτ+∫0t(σt(b(u)-b(uh)),ξt)dτ+∫0t(σ(bu(u)ut-bu(uh)uht),ξt)dτ+∫0t(au(uh)uhtθt,ξt)dτ+∫0t(a(uh)θtt,ξt)dτ+∫0t(bu(uh)uhtθ,ξt)dτ+∫0t(b(uh)θt,ξt)dτ.
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:
(3.12)|∫0t12(au(uh)uhtξt,ξt)dτ|=12|∫0t(au(uh)utξt,ξt)dτ+∫0t12(au(uh)(uht-ut)ξt,ξt)dτ|≤C∫0t‖ξt‖2dτ+C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2+‖ξt‖2)dτ.
For the seventh term on the right-hand side of (3.11), one has
(3.13)|∫0t(bu(uh)uhtξ,ξt)dτ|=|∫0t(bu(uh)(uht-ut)ξ,ξt)+(bu(uh)utξ,ξt)dτ|≤C∫0t(‖ξt‖2+‖ξ‖2)dτ+C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2+‖ξ‖2)dτ.
For the term ∫0t(σt(au(u)ut-au(uh)uht),ξt)dτ on the right side of (3.11), we have
(3.14)|∫0t(σt(au(u)ut-au(uh)uht),ξt)dτ|=|∫0t(σt(au(u)-au(uh))ut+au(uh)(ut-uht),ξt)dτ|≤C∫0t(‖α‖2+‖β‖2+‖αt‖2+‖βt‖2+‖ξt‖2)dτ.
Similarly,
(3.15)|∫0t(σ(bu(u)ut-bu(uh)uht),ξt)dτ|≤∫0t|(σ(bu(u)-bu(uh))ut+bu(uh)(ut-uht),ξt)|dτ≤C∫0t(‖α‖2+‖β‖2+‖αt‖2+‖βt‖2+‖ξt‖2)dτ,|∫0t(au(uh)uhtθt,ξt)dτ|=|∫0t(au(uh)(uht-ut)θt,ξt)dτ+∫0t(au(uh)utθt,ξt)dτ|≤C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2+‖θt‖2)dτ+C∫0t(‖θt‖2+‖ξt‖2)dτ,|∫0t(bu(uh)uhtθ,ξt)dτ|≤∫0t|(bu(uh)(uht-ut)θ,ξt)|dτ+∫0t|(bu(uh)utθ,ξt)|dτ≤C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2+‖θ‖2)dτ+C∫0t(‖θ‖2+‖ξt‖2)dτ.
Inserting (3.12)–(3.15) into (3.11) and using the Cauchy-Schwartz inequality lead to
(3.16)∫0t‖∇⋅ζ‖2dτ+12(a(uh)ξt,ξt)+(b(uh)ξ,ξt)≤C(‖η‖2+‖ξt‖2+‖α‖2+‖β‖2+‖θ‖2+‖θt‖2)+C∫0t(‖θtt‖2+‖θt‖2+‖ξt‖2+‖ηt‖2+‖α‖2+‖β‖2+‖ζ‖2)dτ+C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2+‖ξt‖2)dτ+C‖ξt‖L∞(0,t;L∞)∫0t(‖θt‖2+‖θ‖2+‖ξ‖2)dτ+C∫0t(‖θt‖2+‖θ‖2+‖ξ‖2+‖ξt‖2)dτ.
Integrating (3.16) from 0 to t, using the fact (b(uh)ξ,ξt)=(1/2)(d/dt)(b(uh)ξ,ξ)-(1/2)(bu(uh)uhtξ,ξt) and the inequality
(3.17)∫0t∫0τ|ψ(s)|2dsdτ≤C∫0t|ψ(s)|2ds,
yields
(3.18)‖ξ‖2≤C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2+‖ξt‖2)dτ+C‖ξt‖L∞(0,t;L∞)∫0t(‖θt‖2+‖θ‖2+‖ξ‖2)dτ+C∫0t(‖αt‖2+‖βt‖2+‖α‖2+‖β‖2+‖θt‖2+‖θ‖2+‖θtt‖2+‖ηt‖2+‖ξ‖2+‖ξt‖2+‖ζ‖2)dτ.
Thus, to estimate ∥ξ∥, we need to estimate ∥β∥, ∥βt∥, ∥ζ∥, and ∥ξt∥. Taking vh=β in (3.5) leads to
(3.19)(∇β,∇β)=(ξ,∇β)+(θ,∇β).
By the Cauchy-Schwartz inequality, we obtain
(3.20)‖∇β‖≤C(‖ξ‖+‖θ‖).
Note that β∈Vh⊂H01(Ω) and ∥β∥≤C∥∇β∥. We further have
(3.21)‖β‖≤C(‖ξ‖+‖θ‖).
Differentiating (3.5) with respect to t and choosing vh=βt gives
(3.22)‖∇βt‖≤C(‖ξt‖+‖θt‖).
Similarly, since β∈Vh⊂H01(Ω), one has ∥βt∥≤∥∇βt∥≤C(∥ξt∥+∥θt∥).
Takingwh=ζ in (3.6), one has
(3.23)(ζ,ζ)=(a(uh)ξt,ζ)+(b(uh)ξ,ζ)+(σt(a(u)-a(uh)),ζ)+(σ(b(u)-b(uh)),ζ)+(a(uh)θt,ζ)+(b(uh)θ,ζ)-(η,ζ).
By the Cauchy-Schwartz inequality, we obtain
(3.24)‖ζ‖≤C(‖ξ‖+‖θt‖+‖θ‖+‖η‖+‖α‖+‖β‖+‖ξt‖).
To bound ∥ξt∥2, we differentiate (3.6) with respect to t to obtain
(3.25)(ζt,wh)-(a(uh)ξtt,wh)-(b(uh)ξt,wh)=(au(uh)uhtξt,wh)+(bu(uh)uhtξ,wh)+(σtt(a(u)-a(uh)),wh)+(σt(au(u)ut-au(uh)uht),wh)+(σt(b(u)-b(uh)),wh)+(σ(bu(u)ut-bu(uh)uht),wh)+(a(uh)θtt,wh)+(au(uh)uhtθt,wh)+(b(uh)θt,wh)+(bu(uh)uhtθ,wh)-(ηt,wh),∀wh∈Hh.
Testing (3.25) with wh=ξtt and (3.4) with qh=ζt and combining the resulting equations together lead to
(3.26)(∇⋅ζ,∇⋅ζt)+(a(uh)ξtt,ξtt)+(b(uh)ξt,ξtt)=-(au(uh)uhtξt,ξtt)-(bu(uh)uhtξ,ξtt)-(σtt(a(u)-a(uh)),ξtt)-(σt(au(u)ut-au(uh)uht),ξtt)-(σt(b(u)-b(uh)),ξtt)-(σ(bu(u)ut-bu(uh)uht),ξtt)-(a(uh)θtt,ξtt)-(au(uh)uhtθt,ξtt)-(θt,ζt)-(b(uh)θt,ξtt)-(bu(uh)uhtθ,ξtt)+(ηt,ξtt).
Note that
(3.27)(b(uh)ξt,ξtt)=12ddt(b(uh)ξt,ξt)-12(bu(uh)uhtξ,ξt),(∇⋅ζ,∇⋅ζt)=12ddt(∇⋅ζ,∇⋅ζ),(θt,ζt)=ddt(θt,ζ)-(θtt,ζ).
Thus, (3.26) can be rewritten as
(3.28)12ddt(∇⋅ζ,∇⋅ζ)+(a(uh)ξtt,ξtt)+12ddt(b(uh)ξt,ξt)=-(au(uh)uhtξt,ξtt)-12(bu(uh)uhtξ,ξtt)-(σtt(a(u)-a(uh)),ξtt)-(σt(au(u)ut-au(uh)uht),ξtt)-(σt(b(u)-b(uh),ξtt)-(σ(bu(u)ut-bu(uh)uht),ξtt)-(a(uh)θtt,ξtt)-(au(uh)uhtθt,ξtt)-ddt(θt,ζ)+(θtt,ζ)-(b(uh)θt,ξtt)-(bu(uh)uhtθ,ξtt))+(ηt,ξtt).
Integrating (3.28) from 0 to t yields
(3.29)(∇⋅ζ,∇⋅ζ)+∫0t(a(uh)ξtt,ξtt)+(b(uh)ξt,ξt)=-∫0t(au(uh)uhtξt,ξtt)dτ-12∫0t(bu(uh)uhtξ,ξtt)dτ-∫0t(σtt(a(u)-a(uh)),ξtt)dτ-∫0t(σt(au(u)ut-au(uh)uht),ξtt)dτ-∫0t(σt(b(u)-b(uh)),ξtt)dτ-∫0t(σ(bu(u)ut-bu(uh)uht),ξtt)dτ-∫0t(a(uh)θtt,ξtt)dτ-∫0t(au(uh)uhtθt,ξtt)dτ-(θt,ζ)+∫0t(θtt,ζ)dτ-∫0t(b(uh)θt,ξtt)dτ-∫0t(bu(uh)uhtθ,ξtt)dτ+∫0t(ηt,ξtt)dτ.
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 ɛ>0, it holds that
(3.30)|-∫0t(au(uh)uhtξt,ξtt)dτ|≤|∫0t(au(uh)(uht-ut)ξt,ξtt)dτ|+|∫0t(au(uh)utξt,ξtt)dτ|≤C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2)dτ+C∫0t‖ξt‖2dτ+ɛ(1+‖ξt‖L∞(0,t;L∞))∫0t‖ξtt‖2dτ.
Similarly, we can bound (3.29) as follows:
(3.31)‖∇⋅ζ‖2+‖ξt‖2+∫0t‖ξtt‖2dτ≤C‖ξt‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2)dτ+ɛ(1+‖ξt‖L∞(0,t;L∞))∫0t‖ξtt‖2dτ+C‖ξ‖L∞(0,t;L∞)∫0t(‖αt‖2+‖βt‖2)dτ+ɛ(1+‖ξ‖L∞(0,t;L∞))∫0t‖ξtt‖2dτ+C∫0t(‖ξ‖2+‖ζ‖2+‖θtt‖2+‖θt‖2+‖θ‖2+‖α‖2+‖αt‖2+‖β‖2+‖βt‖2+‖ηt‖2+‖ξt‖2)dτ+‖θt‖‖ζ‖.
In the following error analysis, we make an induction hypothesis:
(3.32)(‖ξt‖L∞(0,t;L∞)+‖ξ‖L∞(0,t;L∞))≤1.
Utilizing (3.32), (3.24), (3.22), (3.21), and Young's inequality, one can reduce (3.31) to
(3.33)‖∇⋅ζ‖2+‖ξt‖2≤C(‖ξ‖2+‖θt‖2+‖θ‖2+‖α‖2+‖η‖2)+C∫0t(‖ξ‖2+‖θtt‖2+‖θt‖2+‖θ‖2+‖α‖2+‖αt‖2+‖ηt‖2+‖ξt‖2)dτ.
Then by Gronwall's inequality, we obtain
(3.34)‖∇⋅ζ‖2+‖ξt‖2≤C(‖ξ‖2+‖θt‖2+‖θ‖2+‖α‖2+‖η‖2)+C∫0t(‖ξ‖2+‖θtt‖2+‖θt‖2+‖θ‖2+‖α‖2+‖αt‖2+‖ηt‖2)dτ.
Furthermore, by (3.24) and (3.34), one has
(3.35)‖ζ‖2≤C(‖ξ‖2+‖θt‖2+‖θ‖2+‖α‖2+‖η‖2)+C∫0t(‖ξ‖2+‖θtt‖2+‖θt‖2+‖θ‖2+‖α‖2+‖αt‖2+‖ηt‖2)dτ.
Therefore, by the estimates of ∥β∥, ∥βt∥, ∥ζ∥, and ∥ξt∥, it follows that
(3.36)‖ξ‖2≤C∫0t(‖ξ‖2+‖θtt‖2+‖θt‖2+‖θ‖2+‖α‖2+‖αt‖2+‖η‖2+‖ηt‖2)dτ.
Applying Gronwall's inequality to the above equation and using the estimates of projection operators give
(3.37)‖ξ‖2≤C∫0t(‖θtt‖2+‖θt‖2+‖θ‖2+‖α‖2+‖αt‖2+‖η‖2+‖ηt‖2)dτ≤Chmin⁡⁡(2k+2,2m+2)(‖ut‖L∞(Hm+1)2+‖u‖L∞(Hm+1)2+‖pt‖L∞(Hk+1)2+‖p‖L∞(Hk+1)2+‖σt‖L∞(Hk+1)2+‖σtt‖L∞(Hk+1)2+‖σ‖L∞(Hk+1)2).
Inserting the estimate of ∥ξ∥ into (3.34) yields
(3.38)‖ξt‖2≤Chmin⁡⁡(2k+2,2m+2).
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, ξt(0)=0. Note that ∥ξ∥L∞(0,t;L∞)+∥ξt∥L∞(0,t;L∞) is continuous w.r.t. t. Then, we conclude that there exists t1∈(0,T] such that
(3.39)‖ξ‖L∞(0,t1;L∞)+‖ξt‖L∞(0,t1;L∞)≤1.
Set t*=supt1. Thus, ∥ξ∥L∞(0,t*;L∞)+∥ξt∥L∞(0,t*;L∞)≤1. Therefore, we have
(3.40)‖ξ(t*)‖+‖ξt(t*)‖≤Chmin⁡⁡(k+1,m+1).
By inverse estimates, we deduce that, for any0≤t≤t*, it holds that
(3.41)‖ξ‖L∞(0,t;L∞)+‖ξt‖L∞(0,t;L∞)≤Chmin⁡⁡(k+1,m+1)-d/2.
Then we can take h>0 sufficiently small such that
(3.42)‖ξ‖L∞(0,t*;L∞)+‖ξt‖L∞(0,t*;L∞)<1.
Again, by the continuity of ∥ξ∥L∞(0,t;L∞)+∥ξt∥L∞(0,t;L∞), we conclude that there exists a positive constant δ such that
(3.43)‖ξ‖L∞(0,t*+δ;L∞)+‖ξt‖L∞(0,t*+δ;L∞)≤1,
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 Theorem3.1 and the standard embedding theorem, we can obtain the L∞ estimate for d=1 and 2 as follows:
(3.44)‖u-uh‖L∞(L∞)≤C2|ln⁡⁡h|d-1hmin⁡⁡(k+1,m+1).


## 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.



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*Source: 570980-2011-10-20.xml*