# A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

Georgoulis, Emmanuil H, Lakkis, Omar and Wihler, Thomas P (2021) A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems. Numerische Mathematik. ISSN 0029-599X

We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an $hp$-version discontinuous Galerkin (DG) time-stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in $\operatorname L_{\infty}(I;\operatorname L2(\Omega))$- and $\operatorname L_{2}(I;\operatorname H^{1}(\Omega))$-type norms when $I$ is the temporal and $\Omega$ the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice to the user. It also exhibits mesh-change estimators in a clear and concise way. We also show how our approach allows the derivation of such bounds in the $\operatorname{H}^1(I;\opeartorname{H}^{-1}(\Omega))$ norm.