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

TL;DR

This work analyzes discontinuous Galerkin time-stepping for nonlinear parabolic equations $u_t=\nabla\cdot f(\nabla u,u)$ under homogeneous Dirichlet conditions, establishing both optimal-order a priori error estimates and conditional a posteriori estimates. The analysis hinges on maximal $L^p$-regularity for non-autonomous linear parabolic problems and a Radau IIA-based interpretation of the dG method, enabling stable, high-order convergence with a reconstruction operator linking the dG scheme to continuous-time formulations. Under precise regularity $u\in W^{q,p}((0,T);W^{2,r}(\Omega)\cap W^{1,r}_0(\Omega))$ with $2/p+d/r<1$, the authors prove $\|u-U\|_{L^p(0,T;W^{2,r}(\Omega))}\le Ck^q$ and analogous bounds for time derivatives and reconstructions, plus a residual-based a posteriori bound $\|u_t-\widehat{U}_t\|_{L^p(L^r)}+\|u-\widehat{U}\|_{L^p(W^{2,r})}\le C\|R\|_{L^p(L^r)}$. The results extend error analysis methods from Runge–Kutta schemes to dG time stepping, providing rigorous foundations for adaptive time discretization of nonlinear parabolic problems. The findings have practical impact for reliable time discretization error control in simulations governed by nonlinear diffusion-type dynamics.

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.

Theorems & Definitions (9)

• Theorem 1.1: A priori error estimates
• Theorem 1.2: Conditional a posteriori error estimate
• Lemma 2.1: KuLL; maximal regularity for non-autonomous equations
• Lemma 2.2: Maximal regularity of Radau IIA methods; KuLL
• Lemma 2.3: Maximal regularity of dG methods in continuous norms
• proof
• Remark 2.1: Equivalence of norms $\| A(\bar{t})\cdot\|_{L^p((0,T);L^r(\varOmega))}$ and $\| A\cdot\|_{L^p((0,T);L^r(\varOmega))}$
• Lemma 3.1: Space-time Sobolev embedding; KuLL
• Lemma 3.2: Schaefer’s fixed point theorem; E