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

TL;DR

This paper proves optimal approximation properties for fully discrete Galerkin solutions of second order parabolic problems, focusing on the gradient in a strong norm, using novel proof techniques that avoid mesh constraints.

Contribution

It introduces a new proof method for error estimates of Galerkin solutions that does not rely on mesh size relationships and applies to interior pointwise approximation.

Findings

Establishes global best approximation in $L^ Infty(I;W^{1, Infty}(\\Omega))$ norm.

Proves interior pointwise best approximation property.

Uses elliptic results and discrete maximal parabolic regularity for proofs.

Abstract

In this paper we establish best approximation property of fully discrete Galerkin solutions of second order 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 spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.