Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-04-30T10:51:57.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Galerkin time-stepping methods for nonlinear parabolic equations

Published online by Cambridge University Press:  15 March 2004

Georgios Akrivis  and
Charalambos Makridakis
Georgios Akrivis
Affiliation:
Computer Science Department, University of Ioannina, 451 10 Ioannina, Greece, akrivis@cs.uoi.gr.
Charalambos Makridakis
Affiliation:
Department of Applied Mathematics, University of Crete, 71409 Heraklion-Crete, Greece, and Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion-Crete, Greece, makr@tem.uoc.gr.
Get access

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.

Keywords

Nonlinear parabolic equationslocal Lipschitz conditioncontinuous and discontinuous Galerkin methodsa priori error analysismonotone operators.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 261 - 289
Copyright
© EDP Sciences, SMAI, 2004

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

Akrivis, G. and Crouzeix, M., Linearly implicit methods for nonlinear parabolic equations. Math. Comp. 73 (2004) 613–635. CrossRef
G. Akrivis and C. Makridakis, Convergence of a time discrete Galerkin method for semilinear parabolic equations. Preprint (2002).
G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep finite element methods for nonlinear parabolic problems. Math. Comp. 67 (1998) 457–477.
G. Akrivis, M. Crouzeix and C. Makridakis, Implicit-explicit multistep methods for quasilinear parabolic equations. Numer. Math. 82 (1999) 521–541.
A.K. Aziz and P. Monk, Continuous finite elements in space and time for the heat equation. Math. Comp. 52 (1989) 255–274.
J.H. Bramble and P.H. Sammon, Efficient higher order single step methods for parabolic problems: Part I, Math. Comp. 35 (1980) 655–677.
G.A. Baker and J. H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations. RAIRO Anal. Numér. 13 (1979) 75–100.
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28 (1991) 43–77.
K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749.
K. Eriksson, C. Johnson and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups. SIAM J. Numer. Anal. 35 (1998) 1315–1325.
D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems. RAIRO Modél. Math. Anal. Numér. 27 (1993) 35–54.
P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15 (1978) 912–928.
C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 107 (1993) 117–129.
C. Johnson, Y.-Y. Nie and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem. SIAM J. Numer. Anal. 27 (1990) 277–291.
C. Johnson and A. Szepessy, Adaptive finite element methods for conservation laws based on a posteriori error estimates. Comm. Pure Appl. Math. 48 (1995) 199–234.
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.
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.
O. Karakashian and C. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations. Math. Comp. (to appear).
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.
C. Makridakis and R.H. Nochetto, A posteriori error estimates for a class of dissipative schemes for nonlinear evolution equations. Preprint (2002).
R.H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp. 69 (2000) 1–24.
R.H. Nochetto, G. Savaré and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math. 53 (2000) 525–589.
A.H. Schatz and L.B. Wahlbin, Interior maximum-norm estimates for finite element methods: Part II. Math. Comp. 64 (1995) 907–928.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997).