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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:[email protected](L. Yi)
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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 

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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