Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem s

TL;DR

This paper establishes maximal parabolic regularity for Galerkin finite element methods applied to non-autonomous linear parabolic equations, providing key estimates for time-discrete and space-time solutions with broad applications.

Contribution

It introduces novel maximal regularity results for the lowest order discontinuous Galerkin method in non-autonomous parabolic problems, including best approximation estimates.

Findings

Maximal parabolic regularity is proven for time semidiscrete solutions.

Space-time fully discrete maximal regularity is established.

Best approximation results in $L^p(0,T;L^2( abla))$ norms are derived.

Abstract

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent coefficients. Such estimates have many applications. As one of the applications we establish best approximations type results with respect to the $L^p(0,T;L^2(\Omega))$ norm for $1\le p\le \infty$.