Error Estimates and Automatic Time Step Control for Nonlinear Parabolic Problems, I

Abstract

In this note we extend results by one of the authors 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 $q \geqq 0$. 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.