Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T03:15:37.306Z Has data issue: false hasContentIssue false
  • English
  • Français

An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems

Part of: Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  10 November 2015

Lijun Yi
Lijun Yi*
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai 200234; and Division of Computational Science, E-institute of Shanghai Universities, Shanghai 200234, China
*
*Corresponding author. Email address:ylj5152@shnu.edu.cn(L. Yi)
Get access

Abstract

The h-p version of the continuous Petrov-Galerkin time stepping method is analyzed for nonlinear initial value problems. An L∞-error bound explicit with respect to the local discretization and regularity parameters is derived. Numerical examples are provided to illustrate the theoretical results.

Keywords

Initial value problemsh-p versiontime stepping methodcontinuous Petrov-Galerkin methoderror bound

MSC classification

Secondary: 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods 65L05: Initial value problems 65L70: Error bounds
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 4 , November 2015 , pp. 301 - 311
Copyright
Copyright © Global-Science Press 2015 

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.)

References

[1]Brunner, H. and Schötzau, D., hp-discontinuous Galerkin time stepping for Volterra integro-differential equations, SIAM J. Num. Anal. 44, 224–245 (2006).Google Scholar
[2]Delfour, M. and Dubeau, F., Discontinuous polynomial approximations in the theory of one-step, hybrid and multistep methods for nonlinear ordinary differential equations, Math. Comp. 47, 169–189 (1986).Google Scholar
[3]Delfour, M., Hager, W. and Trochu, F., Discontinuous Galerkin methods for ordinary differential equations, Math. Comp. 36, 455–473 (1981).Google Scholar
[4]Estep, D. and French, D., Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Modél. Math. Anal. Numér. 28, 815–852 (1994).Google Scholar
[5]Guo, B.Y. and Wang, Z.Q., Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comp. Math. 30, 249–280 (2009).CrossRefGoogle Scholar
[6]Hulme, B.L., One-step piecewise Galerkin methods for initial value problems, Math. Comp. 26, 415–425 (1972).Google Scholar
[7]Hulme, B.L., Discrete Galerkin and related one-step methods for ordinary differential equations, Math. Comp. 26, 881–891 (1972).Google Scholar
[8]Janssen, B. and Wihler, T.P., Existence results for the continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems, arXiv: 1407.5520 (2014).Google Scholar
[9]Mustapha, K., Brunner, H., Mustapha, H. and Schötzau, D., An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type, SIAM J. Num. Anal. 49, 1369–1396 (2011).Google Scholar
[10]Schötzau, D. and Schwab, C., An hp a priori error analysis of the DG time stepping method for initial value problems, Calcolo 37, 207–232 (2000).Google Scholar
[11]Schötzau, D. and Schwab, C., Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method, SIAM J. Num. Anal. 38, 837–875 (2000).Google Scholar
[12]Schwab, C., p- and hp- Finite Element Methods, Oxford University Press, New York (1998).Google Scholar
[13]Wihler, T.P., An a priori error analysis of the hp-version of the continuous Galerkin FEM for nonlinear initial value problems, J. Sci. Comp. 25, 523–549 (2005).Google Scholar
[14]Yi, L.J., An h-p version of the continuous Petrov-Galerkin finite element method for nonlinear Volterra integro-differential equations, J. Sci. Comp. 65, 715–734 (2015).Google Scholar