The variable-order discontinuous Galerkin time stepping scheme for parabolic evolution problems is uniformly $\mathrm{L}^\infty$-stable

TL;DR

This paper proves that a variable-order discontinuous Galerkin time stepping scheme for linear parabolic PDEs maintains uniform $ ext{L}^\infty$-stability, ensuring bounded solutions over time regardless of discretization choices.

Contribution

It establishes the uniform $ ext{L}^\infty$-stability of a variable-order DG time stepping scheme combined with conforming Galerkin spatial discretizations for parabolic problems.

Findings

Global-in-time maximum norm of the discrete solution is bounded by PDE data.

Stability constant is robust and independent of discretization parameters.

Method applies uniformly across different local time steps and approximation orders.

Abstract

In this paper we investigate the $\mathrm{L}^\infty$-stability of fully discrete approximations of abstract linear parabolic partial differential equations. The method under consideration is based on an $hp$-type discontinuous Galerkin time stepping scheme in combination with general conforming Galerkin discretizations in space. Our main result shows that the global-in-time maximum norm of the discrete solution is bounded by the data of the PDE, with a constant that is robust with respect to the discretization parameters (in particular, it is uniformly bounded with respect to the local time steps and approximation orders).