Discrete maximal parabolic regularity for Galerkin finite element methods

TL;DR

This paper establishes maximal parabolic regularity for Galerkin finite element methods, enabling improved error estimates in numerical solutions of linear parabolic equations.

Contribution

It introduces novel maximal regularity results for time discontinuous Galerkin methods, facilitating optimal error analysis without mesh coupling.

Findings

Proves time semidiscrete maximal regularity

Establishes space-time fully discrete regularity

Enables optimal a priori error estimates

Abstract

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They are essential, for example, for establishing optimal a priori error estimates in non- Hilbertian norms without unnatural coupling of spatial mesh sizes with time steps.