Mathematical Problems in Engineering 3 2 . H 1-Galerkin Expanded Mixed Finite Element Discrete Scheme 2 . 1 . Weak Formulation

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.


Introduction
Consider the following nonlinear viscoelasticity-type equation: where Ω is a convex polygonal domain in R 2 with the Lipschitz continuous boundary ∂Ω, J 0, T is the time interval with 0 < T < ∞, and u 0 x and u 1 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 u 0 x , u 1 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 postprocessing 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 H 1 -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 L 2 norm can be obtained if an extra regularity on the solution holds.Recently, H 1 -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 H 1 -Galerkin expanded mixed finite element methods which combines the H 1 -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 H 1 -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 H 1 -Galerkin expanded mixed finite element methods and then discuss the existence and uniqueness of the formulation.In Section 3, we show that the H 1 -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 For theoretical analysis, we also need the following assumptions on the functions involved in problem 1.1 .
Assumption 1.1.1 There exist constants a 1 and a 2 such that 0 < a 1 ≤ a x, u , b x, u ≤ a 2 .
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

Weak Formulation
To define the H 1 -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:

2.2
Then by Green's formula we can further define the following weak formulation of problem 2.2 :

2.3
In order to establish the equivalence between problem 2.2 and the weak form 2. 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 By the divergence theorem 1 , one has 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 Instituting q ψ t into 2.10 and using

2.18
Therefore, 2.10 can equivalently be transformed into the following equation: Recalling Lemma 2.1, one concludes that that is, Combining this with 2.5 and 2.18 results in the desired assertion, and this completes the proof.

Numerical Scheme
Let T h be a quasi-uniform family of subdivision of domain Ω; that is, Ω ∪ K∈T h K with h max {diam K : K ∈ T h }, and let V h be the finite-dimensional subspaces of H 1 0 Ω defined by 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 :

2.24
Let Π h : H → H h denote the Raviart-Thomas interpolation operator 28 which satisfies and the following estimates 26, 28, 29

2.27
With the above notations, the semidiscrete H 1 -Galerkin expanded mixed finite element method for system 2.3 is reduced to find a triple u h , σ h ,

2.28
For the H 1 -Galerkin expanded mixed finite element scheme 2.28 , we claim that there exists a unique solution.
In fact, set V h span {ϕ i } N i 1 and H h span {ψ j } M j 1 .Then σ h , p h ∈ H h and u h ∈ V h , and; hence,

2.30
where 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

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 .

Error Analysis
This section is devoted to the error estimates for the H 1 -Galerkin expanded mixed finite element method.
For error analysis in the following, we need to introduce a projection operator.Let R h : H 1 0 Ω → V h be the Ritz projection defined by Then the following approximation holds 27 :

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:

3.7
where k ≥ 1 and m ≥ 1 for d 2, 3, and the positive constant 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 3.9 Thus, 3.8 can be written as

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:

3.12
For the seventh term on the right-hand side of 3.11 , one has

3.13
For the term

3.15
Inserting 3.12 -3.15 into 3.11 and using the Cauchy-Schwartz inequality lead to

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 ht ξ, ξ t and the inequality

3.19
By the Cauchy-Schwartz inequality, we obtain

3.21
Differentiating 3.5 with respect to t and choosing v h β t gives Taking w h ζ in 3.6 , one has By the Cauchy-Schwartz inequality, we obtain To bound ξ t 2 , we differentiate 3.6 with respect to t to obtain

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

3.26
Note that b u h ξ t , ξ tt 1 2

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

3.30
Similarly, we can bound 3.29 as follows:

3.31
In the following error analysis, we make an induction hypothesis:

3.33
Then by Gronwall's inequality, we obtain

3.34
Furthermore, by 3.24 and 3.34 , one has

Mathematical Problems in Engineering
Therefore, by the estimates of β , β t , ζ , and ξ t , it follows that

3.36
Applying Gronwall's inequality to the above equation and using the estimates of projection operators give

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 t 1 ∈ 0, T such that ξ L ∞ 0,t 1 ;L ∞ ξ t L ∞ 0,t 1 ;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: 3.44

Conclusion
In this paper, H 1 -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 H 1 -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.

Lemma 2.2 see
26 .Let Ω be a bounded domain with a Lipschitz continuous boundary ∂Ω.Then, for any g ∈ L 2 Ω , there exists p ∈ H 1 Ω d ⊂ H div, Ω such that ∇ • p g. Ω is a solution to the system 2.2 if and only if it is a solution to the weak form 2.3 .
a u u u t − a u u h u ht , ξ t dτ on the right side of 3.11 , we have t 0 σ t