Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem

Abstract

This paper is the first part in a series of papers on adaptive finite element methods for parabolic problems. In this paper, an adaptive algorithm is presented and analyzed for choosing the space and time discretization in a finite element method for a linear parabolic problem. The finite element method uses aspace discretization with meshsize variable in space and time and a third-order accurate time discretization with timesteps variable in time. The algorithm is proven to be (i) reliable in the sense that the $L_2 $-error in space is guaranteed to be below a given tolerance for all timesteps and (ii) efficient in the sense that the approximation error is for most timesteps not essentially below the given tolerance. The adaptive algorithm is based on an a posteriors error estimate which proves (i), and sharp a priori error estimates are used to prove (ii). Analogous results are given for the corresponding stationary (elliptic) problem. In the following papers in this series extensions are made, e.g., to timesteps variable also in space and to nonlinear problems.