The variable-order discontinuous Galerkin time stepping scheme for parabolic evolution problems is uniformly L∞-stable

In this paper we investigate the L∞-stability of fully discrete approximations of abstract linear parabolic partial differential equations (PDEs). 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).