Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

HIGHER ORDER FULLY DISCRETE SCHEME COMBINED WITH $H^1$-GALERKIN MIXED FINITE ELEMENT METHOD FOR SEMILINEAR REACTION-DIFFUSION EQUATIONS

  • S. Arul Veda Manickam (Department of Mathematics, industrial Mathematics Group, Indian Institute of Technology Bombay) ;
  • Moudgalya, Nannan-K. (Department of Chem. Engg. Industrial Mathematics Group, Indian Institute of Technology Bomba) ;
  • Pani, Amiya-K. (Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay)
  • Published : 2004.05.01

⟨ Previous Next ⟩

Abstract

We first apply a first order splitting to a semilinear reaction-diffusion equation and then discretize the resulting system by an $H^1$-Galerkin mixed finite element method in space. This semidiscrete method yields a system of differential algebraic equations (DAEs) of index one. A priori error estimates for semidiscrete scheme are derived for both differ-ential as well as algebraic components. For fully discretization, an implicit Runge-Kutta (IRK) methods is applied to the temporal direction and the error estimates are discussed for both components. Finally, we conclude the paper with a numerical example.

Keywords

References

  1. Numer. Math. v.63 Finite difference discretization of the Kuramoto-Sivashinsky equation G. D. Akrivis
  2. Modelisation mathematique et Analyse numerique v.31 Solving the systems of equations arising in the discretization of some nonlinear p. d. e.'s by implicit Rung-Kutta methods G. D. Akrivis;V. A. Dougalis;O. A. Karakashian
  3. Numerical Solution of Initial Value Problems in Differetial-Algebraic Equations K. E. Brenan;S. L. Campbell;L. R. Petzold
  4. J. Austral. Math. Soc. v.3 Coefficients for the study of Runge-Kutta integration processes J. C. Butcher
  5. Numer. Math. v.8 Error estimates for mixed finite element methods for nonlinear parabolic problems S. H. Chou;Q. Li
  6. Lecture Notes in Mathematics v.1409 The Numerical Solution of Differential Algebraic Systems by Rung-Kutta Methods E. Hairer;C. Lubich;M. Roche
  7. Solving Ordinary Differential Equations Ⅱ. Stiff and Differential-Algebraic Problems E. Hairer;G. Wanner
  8. RAIRO Anal. Numer. v.15 Error estimates for some mixed finite element methods for parabolic type problems C. Johnson;V. Thomee
  9. SIAM J. Numer. Anal. v.30 On optimal order error estimates for the nonlinear schrodinger equation O. A. Karakashian;G. D. Akrivis;V. A. Dougalis
  10. Topics in Functional Analysis and Applications S. Kesavan
  11. Ph. D. thesis, Dept. of Mathematics, Indian Institute of Technology Implicit Runge-Kutta methods for differential algebraic equations resulting from the spatial discretization of some nonlinear evolution equations S. A. V. Manickam
  12. SIAM J. Numer. Anal. v.35 An $H^1$-Galerkin mixed finite element method for the parpabolic partial differential equations A. K. Pani
  13. SIAM J. Numer. Anal. v.23 Order results for implicit Rung-Kutta methods applied to differential/algebraic systems L. R. Petzold
  14. SIAM J. Numer. Anal. v.12 On Galerkin methods in semilinear parabolic problems V. Thomee;L. Wahlbin
  15. SIAM J. Numer. Anal. v.10 A priori $L^2$-error estimates for Galerkin approximations to parabolic partial differential equations M. F. Wheeler