Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis; Charalambos Makridakis

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 2, page 261-289
  • ISSN: 0764-583X

Abstract

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We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

How to cite

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Akrivis, Georgios, and Makridakis, Charalambos. "Galerkin time-stepping methods for nonlinear parabolic equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.2 (2004): 261-289. <http://eudml.org/doc/245171>.

@article{Akrivis2004,
abstract = {We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.},
author = {Akrivis, Georgios, Makridakis, Charalambos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {nonlinear parabolic equations; local Lipschitz condition; continuous and discontinuous Galerkin methods; a priori error analysis; monotone operators; discontinuous and continuous Galerkin methods; space-time finite element; time discretization; numerical examples},
language = {eng},
number = {2},
pages = {261-289},
publisher = {EDP-Sciences},
title = {Galerkin time-stepping methods for nonlinear parabolic equations},
url = {http://eudml.org/doc/245171},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Akrivis, Georgios
AU - Makridakis, Charalambos
TI - Galerkin time-stepping methods for nonlinear parabolic equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 2
SP - 261
EP - 289
AB - We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.
LA - eng
KW - nonlinear parabolic equations; local Lipschitz condition; continuous and discontinuous Galerkin methods; a priori error analysis; monotone operators; discontinuous and continuous Galerkin methods; space-time finite element; time discretization; numerical examples
UR - http://eudml.org/doc/245171
ER -

References

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  2. [2] G. Akrivis and C. Makridakis, Convergence of a time discrete Galerkin method for semilinear parabolic equations. Preprint (2002). 
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  12. [12] O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method. Math. Comp. 67 (1998) 479–499. Zbl0896.65068
  13. [13] O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal. 36 (1999) 1779–1807. Zbl0934.65110
  14. [14] O. Karakashian and C. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations. Math. Comp. (to appear). Zbl1057.65066MR2085403
  15. [15] C. Makridakis and I. Babuška, On the stability of the discontinuous Galerkin method for the heat equation. SIAM J. Numer. Anal. 34 (1997) 389–401. Zbl0873.65095
  16. [16] C. Makridakis and R.H. Nochetto, A posteriori error estimates for a class of dissipative schemes for nonlinear evolution equations. Preprint (2002). 
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