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Discrete maximal regularity for the discontinuous Galerkin time-stepping method without logarithmic factor

Takahito Kashiwabara, Tomoya Kemmochi

TL;DR

The paper delivers a logarithmic-factor-free discrete maximal-regularity estimate for the discontinuous Galerkin time-stepping scheme applied to parabolic-type problems in a reflexive Banach space. Central to the approach is the temporally regularized Green's function, which converts the discrete maximal-regularity problem into weighted interpolation-error estimates on a quasi-uniform grid. The authors establish key bounds for the time derivative and jumps, reduce the core analysis to estimates on $Au_\tau$, and verify the result through a detailed treatment of initial-value problems and a contour-based decomposition. The results extend to one-step versions, yield optimal-order error estimates, and apply to fully discrete settings via discrete elliptic operators, providing a solid foundation for robust DG approximations of nonlinear parabolic problems.

Abstract

Maximal regularity is a kind of a priori estimates for parabolic-type equations and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.

Discrete maximal regularity for the discontinuous Galerkin time-stepping method without logarithmic factor

TL;DR

The paper delivers a logarithmic-factor-free discrete maximal-regularity estimate for the discontinuous Galerkin time-stepping scheme applied to parabolic-type problems in a reflexive Banach space. Central to the approach is the temporally regularized Green's function, which converts the discrete maximal-regularity problem into weighted interpolation-error estimates on a quasi-uniform grid. The authors establish key bounds for the time derivative and jumps, reduce the core analysis to estimates on , and verify the result through a detailed treatment of initial-value problems and a contour-based decomposition. The results extend to one-step versions, yield optimal-order error estimates, and apply to fully discrete settings via discrete elliptic operators, providing a solid foundation for robust DG approximations of nonlinear parabolic problems.

Abstract

Maximal regularity is a kind of a priori estimates for parabolic-type equations and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.
Paper Structure (15 sections, 14 theorems, 149 equations)

This paper contains 15 sections, 14 theorems, 149 equations.

Table of Contents

  1. Introduction and results
  2. Preliminaries
  3. Properties of the DG time-stepping method
  4. Interpolation operator
  5. The estimates for time derivative and jump
  6. Temporally regularized Green's function
  7. Temporally regularized delta and Green's functions
  8. Reduction to weighted estimates for regularized Green's function
  9. Weighted interpolation error estimate
  10. Proof of discrete maximal regularity
  11. Proof of Proposition 3.1
  12. Estimates for the initial value problem
  13. Corollaries
  14. One-step version of discrete maximal regularity
  15. Optimal order error estimate

Key Result

Theorem 1.1

Let $X$ be a reflexive Banach space, $A$ be a densely defined closed linear operator on $X$, $p \in (1,\infty)$, $f \in L^p(J;X)$, and $u_0 \in {(X,D(A))_{1-\frac{1}{p},p}}$. Assume that $0 \in \rho(A)$, $A$ has maximal regularity, the family of meshes $\{ \mathcal{J}_\tau \}_\tau$ is quasi-uniform, where $C$ is independent of $\tau$, $f$, and $u_0$.

Theorems & Definitions (27)

  • Theorem 1.1: Discrete maximal regularity
  • Lemma 2.1
  • Corollary 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 17 more