Convergence in $\ell_2$ and $\ell_\infty$ Norm of One-Stage AMF-W-Methods for Parabolic Problems

Abstract

For the numerical solution of parabolic problems with a linear diffusion term, linearly implicit time integrators are considered. To reduce the cost on the linear algebra level, an alternating direction implicit approach is applied (so-called AMF-W-methods). The present work proves optimal bounds of the global error for two classes of $1$-stage methods in the Euclidean $\ell_2$ norm as well as in the maximum norm $\ell_\infty$. The bounds are valid under a very weak step size restriction that covers PDE convergence, where the time step size is of the same order as the spatial grid size.