A priori and a posteriori error estimates for discontinuous Galerkin time-discrete methods via maximal regularity

TL;DR

This paper develops optimal a priori and a posteriori error estimates for discontinuous Galerkin time-discrete methods applied to linear parabolic equations, using maximal regularity in UMD Banach spaces, with implications for nonlinear problems.

Contribution

It introduces a variational analysis approach for error estimation that extends to nonlinear parabolic equations, improving upon previous consistency-based methods.

Findings

Optimal error estimates achieved under minimal regularity assumptions.

Method applies to both autonomous and nonautonomous linear equations.

Framework extends to nonlinear parabolic problems.

Abstract

The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations; in contrast to the earlier work. Both autonomous and nonautonomous linear equations are considered.