Mathematical Problems in Engineering

VolumeΒ 2011Β (2011), Article IDΒ 570980, 18 pages

http://dx.doi.org/10.1155/2011/570980

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

^{1}School of Management Science, Qufu Normal University, Rizhao, Shandong 276800, China^{2}School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261000, China^{3}School 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 -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.

#### 1. Introduction

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 [1β4], and the problem has a unique sufficiently smooth solution with the regularity condition provided that the given data , , , , and 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 (). 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 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 [10β16]. 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 [17] 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 [18β22]. 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 -Galerkin expanded mixed finite element methods which combines the -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 -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 a_{1} and a_{2} 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

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 be a bounded domain with a Lipschitz continuous boundary . Then, for any , there exists and divergence free such that and .*

Lemma 2.2 (see [26]). * 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.

Theorem 2.3. * Under the conditions of Lemmas 2.1 and 2.2, 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 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 [1], 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]:

Let denote the Raviart-Thomas interpolation operator [28] which satisfies and the following estimates [26, 28, 29]

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 [27]:

Let Utilizing (2.3), (2.28), and auxiliary projections (3.1), (2.25), we can obtain the following error equations:

Theorem 3.1. *Let and be the solutions to (2.3) and (2.28), respectively. Then the following error estimates hold:
**
where and for , and the positive constant depends on , , , , , , .*

*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)β(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:

#### 4. Conclusion

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.

#### 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).

#### References

- G. Arfken,
*Mathematical Methods for Physicists*, Academic Press, New York, NY, USA, 3rd edition, 1985. View at Zentralblatt MATH - A. C. Eringen,
*Nonlinear Theory of Continuous Media*, McGraw-Hill Book, New York, NY, USA, 1962. - N. Witold,
*Dynamics of Elastic Systems*, John Wiley & Sons, New York, NY, USA, 1963. - B. D. Coleman, βThermodynamics of materials with memory,β
*Archive for Rational Mechanics and Analysis*, vol. 17, pp. 1β16, 1964. View at Google Scholar - M. R. Ohm, H. Y. Lee, and J. Y. Shin, βError estimate of fully discrete discontinuous Galerkin approximation for the viscoelasticity type equation,β
*Journal of Applied Mathematics and Computing*, vol. 37, pp. 473β495, 2011. View at Publisher Β· View at Google Scholar - Y. P. Lin, V. ThomΓ©e, and L. B. Wahlbin, βRitz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations,β
*SIAM Journal on Numerical Analysis*, vol. 28, no. 4, pp. 1047β1070, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Q. Lin and S. Zhang, βA direct global superconvergence analysis for Sobolev and viscoelasticity type equations,β
*Applications of Mathematics*, vol. 42, no. 1, pp. 23β34, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Jin, T. Liu, and S. Zhang, βGlobal superconvergence analysis of Wilson element for Sobolev and viscoelasticity type equations,β
*Journal of Systems Science and Complexity*, vol. 17, no. 4, pp. 452β463, 2004. View at Google Scholar Β· View at Zentralblatt MATH - D. Shi, Y. Peng, and S. Chen, βSuperconvergence of a nonconforming finite element approximation to viscoelasticity type equations on anisotropic meshes,β
*Numerical Mathematics*, vol. 15, no. 4, pp. 375β384, 2006. View at Google Scholar Β· View at Zentralblatt MATH - P. A. Raviart and J. M. Thomas, βA mixed finite element method for 2nd order elliptic problems,β in
*Mathematical Aspects of Finite Element Methods*, vol. 606 of*Lecture Notes in Mathematics*, pp. 292β315, 1977. View at Google Scholar Β· View at Zentralblatt MATH - D. N. Arnold, F. Brezzi, and D. Jim Jr., βA new mixed finite element for plane elasticity,β
*Japan Journal of Industrial and Applied Mathematics*, vol. 1, no. 2, pp. 347β367, 1984. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. E. Ewing and R. D. Lazarov, βSuperconvergence of the velocity along the Gauss lines in mixed finite element methods,β
*SIAM Journal on Numerical Analysis*, vol. 28, no. 4, pp. 1015β1029, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Douglas Jr., R. E. Ewing, and M. F. Wheeler, βA time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media,β
*RAIRO Analyse NumΓ©rique*, vol. 17, no. 3, pp. 249β265, 1983. View at Google Scholar Β· View at Zentralblatt MATH - J. Douglas Jr., R. E. Ewing, and M. F. Wheeler, βThe approximation of the pressure by a mixed method in the simulation of miscible displacement,β
*RAIRO Analyse NumΓ©rique*, vol. 17, no. 1, pp. 17β33, 1983. View at Google Scholar Β· View at Zentralblatt MATH - T. J. Sun and D. P. Yang, βThe finite difference streamline diffusion methods for Sobolev equations with convection-dominated term,β
*Applied Mathematics and Computation*, vol. 125, no. 2-3, pp. 325β345, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Auricchio, F. Brezzi, and C. Lovadina,
*Mixed Finite Element Methods*, John Wiley & Sons, New York, NY, USA, 2004. - A. K. Pani, βAn H
^{1}-Galerkin mixed finite element method for parabolic difference equations,β*SIAM Journal on Numerical Analysis*, vol. 35, no. 2, pp. 712β727, 1998. View at Publisher Β· View at Google Scholar - A. K. Pani and G. Fairweather, βH
^{1}-Galerkin mixed finite element methods for parabolic partial integro-differential equations,β*IMA Journal of Numerical Analysis*, vol. 22, no. 2, pp. 231β252, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Zhou, βAn H
^{1}-Galerkin mixed finite element method for a class of heat transport equations,β*Applied Mathematical Modelling*, vol. 34, no. 9, pp. 2414β2425, 2010. View at Publisher Β· View at Google Scholar - Y. Liu and H. Li, βH
^{1}-Galerkin mixed finite element methods for pseudo-hyperbolic equations,β*Applied Mathematics and Computation*, vol. 212, no. 2, pp. 446β457, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - L. Guo and H. Z. Chen, βH
^{1}-Galerkin mixed finite element method for the Sobolev equation,β*Journal of Systems Science and Mathematical Sciences*, vol. 26, no. 3, pp. 301β314, 2006. View at Google Scholar Β· View at Zentralblatt MATH - A. K. Pani, R. K. Sinha, and A. K. Otta, βAn H
^{1}-Galerkin mixed method for second order hyperbolic equations,β*International Journal of Numerical Analysis and Modeling*, vol. 1, no. 2, pp. 111β139, 2004. View at Google Scholar - H. Z. Chen and H. Wang, βAn optimal-order error estimate on an H
^{1}-Galerkin mixed method for a nonlinear parabolic equation in porous medium flow,β*Numerical Methods for Partial Differential Equations*, vol. 26, no. 1, pp. 188β205, 2010. View at Publisher Β· View at Google Scholar - Z. Chen, βExpanded mixed finite element methods for linear second-order elliptic problems. I,β
*RAIRO ModΓ©lisation MathΓ©matique et Analyse NumΓ©rique*, vol. 32, no. 4, pp. 479β499, 1998. View at Google Scholar Β· View at Zentralblatt MATH - F. Brezzi and M. Fortin,
*Mixed and Hybrid Finite Element Methods*, Springer, New York, NY, USA, 1991. - S. C. Brenner and L. R. Scott,
*The Mathematical Theory of Theory of Finite Element Methods*, Springer, New York, NY, USA, 1991. - P. G. Ciarlet,
*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, The Netherlands, 1978. - J. Douglas Jr. and J. E. Roberts, βGlobal estimates for mixed methods for second order elliptic equations,β
*Mathematics of Computation*, vol. 44, no. 169, pp. 39β52, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Brezzi, βOn the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,β
*RAIRO Analyse NumΓ©rique*, vol. 8, pp. 129β151, 1974. View at Google Scholar Β· View at Zentralblatt MATH - J. Douglas Jr., βSuperconvergence in the pressure in the simulation of miscible displacement,β
*SIAM Journal on Numerical Analysis*, vol. 22, no. 5, pp. 962β969, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH