Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic problems

Dmitriy Leykekhman, Boris Vexler

Published 2016-06-20Version 1

In this paper we establish best approximation property of fully discrete Galerkin solutions of the second parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in \cite{LeykekhmanD_VexlerB_2016b}. In addition, the proof does not require any relationship between spacial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.