Error analysis for discontinuous Galerkin time-stepping methods for nonlinear parabolic equations via maximal regularity

TL;DR

This paper analyzes the error behavior of discontinuous Galerkin time-stepping methods applied to nonlinear parabolic equations, providing new a priori and a posteriori estimates based on maximal regularity.

Contribution

It introduces a novel error analysis framework for DG methods on nonlinear parabolic equations using maximal regularity, extending previous linear case results.

Findings

Established a priori error estimates for DG methods

Derived conditional a posteriori error estimates

Extended maximal regularity techniques to nonlinear problems

Abstract

We consider the discretization of a class of nonlinear parabolic equations by discontinuous Galerkin time-stepping methods and establish a priori as well as conditional a posteriori error estimates. Our approach is motivated by the error analysis in [9] for Runge-Kutta methods for nonlinear parabolic equations; in analogy to [9], the proofs are based on maximal regularity properties of discontinuous Galerkin methods for non-autonomous linear parabolic equations.