Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T09:14:44.238Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

Published online by Cambridge University Press:  03 June 2015

Luoping Chen and
Yanping Chen
Luoping Chen*
Affiliation:
School of Mathematic, Southwest Jiaotong University, Chengdu 611756, P.R. China
Yanping Chen*
Affiliation:
School of Mathematic science, South China Normal University, Guangzhou 510631, P.R. China
*
Email: clpchenluoping@163.com
Corresponding author. Email: yanpingchen@scnu.edu.cn
Get access

Abstract

In this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h1/2). As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

Keywords

65M1265M1565M60Two-grid methodreaction-diffusion equationsmixed finite element methods

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 2 , April 2014, pp. 203 - 219
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor cofficients as cell-centered finite differences. SIAM J. Numer. Anal. 32,828–852 (1997).CrossRefGoogle Scholar
[2]Brezzi, F., Douglas, J. Jr., Marini, L.D.: Two families of mixed finite elemetns for second order elliptic problems. Numer. Math. 47, 217–235 (1985).Google Scholar
[3]Chen, L., Chen, Y.: Two-grid method for nonlinear reaction diffusion equations by mixed finite element methods. J. Sci. Comp. 49(3), 383–401(2011).Google Scholar
[4]Chen, Y., Huang, Y.: A multilevel method for finite element solutions for singular two-point boundary value problems. Natural Science Journal of Xiangtan University, 16, 23–26 (1994) (in chinese).Google Scholar
[5]Chen, Y., Huang, Y., Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations. Int. J. Numer. Meth. Engng. 57,193–209 (2003).Google Scholar
[6]Chen, Y., Yu, D., Liu, S.: Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods. Int. J. Numer. Meth. Engng. 69, 408–422 (2007).Google Scholar
[7]Douglas, J. Jr., Ewing, R.E., Wheeler, M.F.: The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO: Analyse Numerique. 17, 17–33(1983).Google Scholar
[8]Dawson, C.N., Wheeler, M.F.: Two-grid method for mixed finite element approximations of non-linear parabolic equations. Contemporary mathematics. 180,191–203 (1994).Google Scholar
[9]Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference sceme for nonlinear parabolic euqations. SIAMJ. Numer. Anal. 35, 435–452 (1998).CrossRefGoogle Scholar
[10]Garcia, M.F.: Improved error estimates for mixed finite element approximations for nonlinear parabolic equations: the continuously-time case. Numerical methods for partial diferential equations, 10,129–149 (1994).Google Scholar
[11]Huyakorn, P.S., Pinder, G.F.: Computational methods in Surface Flow. Academic Press: New York(1983).Google Scholar
[12]Layton, W., Meir, A.J., Schmidt, P.G. A two-level discretization method for the stationary MHd equations, Electronic Transactions on Numerical Analysis. 6(12),198–210(1997).Google Scholar
[13]Murray, J.: Mathmatical Biology, 2nd edn. Springer: New York(1993).CrossRefGoogle Scholar
[14]Squeff, M.C.J.: Superconvergence of mixed finite elements for parabolic equations. Math. Model. Numer. Anal. 21,327–352 (1987).Google Scholar
[15]Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems, Math Aspects of the Finite Element Method. Lecture Notes in Math, 606. Springer, Berlin, 292–315 (1977).Google Scholar
[16]Thomée, V.: Galerkin finite element methods for parabolic problems, Springer-Verlag(1997).Google Scholar
[17]Wu, L., Allen, M.B.: A two-grid method for mixed finite element solution of reaction-diffusion equations. Numerical Methods for Partial differential Equations. 15, 317–332 (1999).Google Scholar
[18]Xu, J.: Two-grid discretization techniques for linear and non-linear PDEs, SIAM J. Numer. Anal. 33,1759–1777 (1996).Google Scholar
[19]Xu, J.: A novel two-grid method for semi-linear equations. SIAM J. Sci. Comput. 15,231–237 (1994).Google Scholar
[20]Xu., J, Zhou, A.: A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70(233),17–25(1999).Google Scholar
[21]Zhong, L., Shu, S., Wang, J., Xu, J., Two-grid methods for time-harmonic Maxwell equations,Numerical Linear Algebra with Applications. 1,93–111(2012).Google Scholar