The variable-order discontinuous Galerkin time stepping scheme for parabolic evolution problems is uniformly $\mathrm{L}^\infty$-stable
Provides a theoretical stability guarantee for a numerical method used in solving parabolic evolution problems.
The paper proves that the hp-type discontinuous Galerkin time stepping scheme for abstract linear parabolic PDEs is uniformly L∞-stable, with a constant robust to discretization parameters.
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).