Research Article | Open Access
Haitao Che, Yiju Wang, Zhaojie Zhou, "An Optimal Error Estimates of H1-Galerkin Expanded Mixed Finite Element Methods for Nonlinear Viscoelasticity-Type Equation", Mathematical Problems in Engineering, vol. 2011, Article ID 570980, 18 pages, 2011. https://doi.org/10.1155/2011/570980
An Optimal Error Estimates of H1-Galerkin Expanded Mixed Finite Element Methods for Nonlinear Viscoelasticity-Type Equation
We investigate a -Galerkin mixed finite element method for nonlinear viscoelasticity equations based on -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.
Consider the following nonlinear viscoelasticity-type equation: where is a convex polygonal domain in with the Lipschitz continuous boundary , is the time interval with , and and are, respectively, the initial data functions defined on . 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 , , , , and are sufficiently smooth .
For problem (1.1), by adopting finite element method, Lin et al.  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 (). Latter, Lin and Zhang  presented a direct analysis for global superconvergence for this problem without using the Ritz projection or its modified forms. Jin et al.  and Shi et al.  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 by the unknown scalar 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 and its flux 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  proposed a new mixed finite element method, called the -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 norm can be obtained if an extra regularity on the solution holds. Recently, -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  proposed -Galerkin expanded mixed finite element methods which combines the -Galerkin formulation and the expanded mixed finite element methods  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 -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 -Galerkin expanded mixed finite element methods and then discuss the existence and uniqueness of the formulation. In Section 3, we show that the -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 to denote the space with norm and . 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 .
(2) The functions , and are Lipschitz continuous with respect to , and there exists such that .
2. -Galerkin Expanded Mixed Finite Element Discrete Scheme
2.1. Weak Formulation
To define the -Galerkin expanded mixed finite element procedure, we introduce vector and split (1.1) into a first-order system as follows: Then by Green's formula we can further define the following weak formulation of problem (2.2): find such that
Lemma 2.1 (see ). Let be a bounded domain with a Lipschitz continuous boundary . Then, for any , there exists and divergence free such that and .
Lemma 2.2 (see ). Let be a bounded domain with a Lipschitz continuous boundary . Then, for any , there exists such that .
Now we are in a position to state our main result in this subsection.
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 in the third equation of (2.3) leads to which implies
By Lemma 2.1, there exist and divergence free such that and . Choosing in the second equation of (2.3) yields By the divergence theorem , one has Substituting (2.7) into (2.6) yields which means that Inserting (2.5) and (2.9) into the first equation of (2.2) and applying the divergence theorem to the first term, for any , one has Instituting into (2.10) and using lead to Integrating from 0 to with respect to time results in Differentiating (2.9) with respect to , one obtains By the fifth equation in (2.3), we deduce that which implies Integrating the equation with respect to from 0 to gives By (2.9) and the forth equation in (2.2), we deduce which leads to Therefore, (2.10) can equivalently be transformed into the following equation: For , by Lemma 2.2, there exists such that . Thus, (2.19) reduces to 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.
2.2. Numerical Scheme
Let be a quasi-uniform family of subdivision of domain ; that is, with = max , and let be the finite-dimensional subspaces of defined by where denotes the space of polynomials of degree at most on . Moreover, we denote the vector space in mixed finite element spaces with index by . It is well known that both and satisfy the inverse property and the following approximation properties [26, 27]:
With the above notations, the semidiscrete -Galerkin expanded mixed finite element method for system (2.3) is reduced to find a triple such that
For the -Galerkin expanded mixed finite element scheme (2.28), we claim that there exists a unique solution.
In fact, set and . Then and , and; hence, Taking , , , , in (2.28) leads to where and , are given.
Note that matrix in (2.31) is positive definite. Thus, by the third equation in (2.30), one has Inserting the above equality into the first equation of (2.30) yields By the standard arguments on the initial-value problem of a system of ordinary differential equations, we can obtain existence and uniqueness of . The existence and uniqueness of and follow from the existence and uniqueness of .
3. Error Analysis
This section is devoted to the error estimates for the -Galerkin expanded mixed finite element method.
For error analysis in the following, we need to introduce a projection operator. Let be the Ritz projection defined by Then the following approximation holds :
Proof. Since estimates of , , and can be obtained by (3.2) and (2.26), it suffices to estimate , , and .
Instituting into (3.6) and in (3.4) gives It is easy to check that Thus, (3.8) can be written as Integrating this system from 0 to yields 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: For the seventh term on the right-hand side of (3.11), one has For the term on the right side of (3.11), we have Similarly,
Inserting (3.12)–(3.15) into (3.11) and using the Cauchy-Schwartz inequality lead to Integrating (3.16) from 0 to , using the fact and the inequality yields Thus, to estimate , we need to estimate , , , and . Taking in (3.5) leads to By the Cauchy-Schwartz inequality, we obtain Note that and . We further have Differentiating (3.5) with respect to and choosing gives Similarly, since , one has .
Taking in (3.6), one has By the Cauchy-Schwartz inequality, we obtain To bound , we differentiate (3.6) with respect to to obtain Testing (3.25) with and (3.4) with and combining the resulting equations together lead to Note that Thus, (3.26) can be rewritten as Integrating (3.28) from 0 to yields
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 , it holds that Similarly, we can bound (3.29) as follows: In the following error analysis, we make an induction hypothesis: Utilizing (3.32), (3.24), (3.22), (3.21), and Young's inequality, one can reduce (3.31) to Then by Gronwall's inequality, we obtain Furthermore, by (3.24) and (3.34), one has Therefore, by the estimates of , , , and , it follows that Applying Gronwall's inequality to the above equation and using the estimates of projection operators give Inserting the estimate of into (3.34) yields Thus, the estimates of and follow from the estimate of .
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 , then , . Note that is continuous w.r.t. . Then, we conclude that there exists such that Set . Thus, . Therefore, we have
By inverse estimates, we deduce that, for any , it holds that Then we can take sufficiently small such that Again, by the continuity of , we conclude that there exists a positive constant such that which contracts to the definition of . 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 estimate for and 2 as follows:
In this paper, -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 -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.
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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