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The discontinuous Galerkin method for semilinear parabolic problems. (English) Zbl 0768.65065
The authors propose a Galerkin scheme for an initial-boundary value problem corresponding to a semilinear (a smooth nonlinearity refers only to the unknown function and that does not affect its derivatives) parabolic equation.
The use finite dimensional spaces with basis functions that are continuous in the space variables and discontinuous in time. They allow variable spatial meshes (that will change on each level time) and time steps.
In fact they extend some earlier results of K. Eriksson and C. Johnson [SIAM J. Numer. Anal. 28, 43–77 (1991; Zbl 0732.65093)] to such nonlinear problems taking into account, essentially, the effect of numerical integration on a priori error analysis.
A numerical example for a polynomial nonlinearity is carried out in the last section.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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References:
[1] T. DUPONT, Mesh modification for evolution equations, Math. Comp. 39 (1982), 85-107. Zbl0493.65044 MR658215 · Zbl 0493.65044 · doi:10.2307/2007621
[2] 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. Zbl0732.65093 MR1083324 · Zbl 0732.65093 · doi:10.1137/0728003
[3] K. ERIKSSON, C. JOHNSON and V. THOMÉE, Time discretization of parabolic problems by the discontinuous Galerkin method, M2AN 19 (1985), 611-643. Zbl0589.65070 MR826227 · Zbl 0589.65070 · eudml:193462
[4] Y.-Y. NIE and V. THOMÉE, A lumped mass finite-element method with quadrature for a non-linear parabolic problem, IMA J. Numer. Anal. 5, 371-396. Zbl0591.65079 MR816063 · Zbl 0591.65079 · doi:10.1093/imanum/5.4.371
[5] V. THOMÉE, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, 1984. Zbl0528.65052 MR744045 · Zbl 0528.65052 · doi:10.1007/BFb0071790
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