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In this paper we investigate the $\mathrm{L}^\infty$-stability of fullydiscrete approximations of abstract linear parabolic partial differentialequations. The method under consideration is based on an $hp$-typediscontinuous Galerkin time stepping scheme in combination with generalconforming Galerkin discretizations in space. Our main result shows that theglobal-in-time maximum norm of the discrete solution is bounded by the data ofthe PDE, with a constant that is robust with respect to the discretizationparameters (in particular, it is uniformly bounded with respect to the localtime steps and approximation orders).

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