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Error estimates and automatic time step control for nonlinear parabolic problems. I. (English) Zbl 0618.65104

We extend results by the second author on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem. The method for time discretization is the so-called discontinuous Galerkin method based on using piecewise polynomials of degree \(q\geq 0\). We consider in this note the case \(q=0\) corresponding to a variant of the backward Euler method. We prove a new almost optimal error estimate and present a related new algorithm for automatic time step control. This algorithm is very simple but yet is efficient and gives control of the global error.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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