On positivity and maximum-norm contractivity in time stepping methods for parabolic equations

Journal article, 2010

In an earlier paper the last two authors studied spatially semidiscrete piecewise linear finite element approximations of the heat equation and showed that, in the case of the standard Galerkin method, the solution operator of the initial-value problem is neither positive nor contractive in the maximum-norm for small time, but that for the lumped mass method these properties hold, if the triangulations are essentially of Delaunay type. In this paper we continue the study by considering fully discrete analogues obtained by discretization also in time. The above properties then carry over to the backward Euler time stepping method, but for other methods the results are more restrictive. We discuss in particular the θ-method and the (0; 2) Padé approximation in one space dimension.