Pointwise best approximation results for Galerkin finite element solutions of parabolic problems

TL;DR

This paper proves a new best approximation property for fully discrete Galerkin finite element solutions of second order parabolic problems in the $L^ abla$ norm, applicable in three dimensions and with arbitrary time discretization order.

Contribution

It introduces a novel proof technique that achieves best approximation results in three space dimensions without mesh size restrictions, using elliptic and resolvent estimates.

Findings

First best approximation results in 3D for parabolic problems

Applicable to arbitrary order discontinuous Galerkin methods in time

No mesh size and time step relationship required

Abstract

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method uses of continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [16]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish a local best approximation property that shows a more local behavior of the error at a given point.