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

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 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.