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

Georgios Akrivis, Stig Larsson

TL;DR

The paper analyzes discontinuous Galerkin time discretizations for linear parabolic equations with maximal $L^p$-regularity in UMD Banach spaces. It develops a variational framework combined with a reconstruction operator to obtain optimal-order a priori and a posteriori error estimates, for both autonomous and nonautonomous problems, and shows how these results extend to nonlinear parabolic equations. Central to the approach is maximal parabolic regularity, which enables precise control of time derivatives and the $A$-operator, yielding bounds of the form $\|A(u-U)\|_{L^p}$ and residual-based estimators with the optimal decay $k^{q}$. The work provides a robust, operator-theoretic foundation for adaptive time stepping and extends prior Radau IIA-based analyses by leveraging variational techniques and a unified nonautonomous treatment.

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.

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

TL;DR

The paper analyzes discontinuous Galerkin time discretizations for linear parabolic equations with maximal -regularity in UMD Banach spaces. It develops a variational framework combined with a reconstruction operator to obtain optimal-order a priori and a posteriori error estimates, for both autonomous and nonautonomous problems, and shows how these results extend to nonlinear parabolic equations. Central to the approach is maximal parabolic regularity, which enables precise control of time derivatives and the -operator, yielding bounds of the form and residual-based estimators with the optimal decay . The work provides a robust, operator-theoretic foundation for adaptive time stepping and extends prior Radau IIA-based analyses by leveraging variational techniques and a unified nonautonomous treatment.

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.
Paper Structure (18 sections, 4 theorems, 89 equations)

This paper contains 18 sections, 4 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Maximal parabolic regularity
  3. The numerical methods
  4. A reconstruction operator
  5. Main results
  6. A priori error estimates
  7. A discrete $\ell^p(X)$-norm and its equivalence to the continuous $L^p(X)$-norm
  8. Maximal parabolic regularity of the dG method
  9. An interpolant and its approximation properties
  10. A priori error estimates
  11. Extension to nonautonomous equations
  12. Maximal parabolic regularity
  13. A priori error estimates
  14. A posteriori error estimates
  15. Interpolation error estimates
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $p\in(1,\infty)$ and assume that the solution of ivp is sufficiently regular, $u\in W^{q,p} ((0,T);{\mathscr{D}}(A) ).$ Then, the dG approximation $U\in {\mathcal{V}}_k^{\text{d}} (q-1)$ satisfies the estimate Furthermore, if $u_0\in {\mathscr{D}}(A)$ and $u\in W^{q+1,p} ((0,T);X ),$ for the reconstruction $\widehat{U}\in {\mathcal{V}}_k^{\text{c}} (q),$ we have The constant $C$ depends on $

Theorems & Definitions (7)

  • Theorem 1.1: A priori error estimates
  • Theorem 1.2: A posteriori error estimate
  • proof
  • Theorem 3.1: A priori error estimates
  • proof
  • Lemma A.1: Reproduction property
  • proof