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We consider fully discrete time-space approximations of abstract linearparabolic partial differential equations (PDEs) consisting of an $hp$-versiondiscontinuous Galerkin (DG) time stepping scheme in conjunction with standard(conforming) Galerkin discretizations in space. We derive abstract computable aposteriori error bounds resulting, for instance, in concrete bounds in$L_{\infty}(I;L_2(\Omega))$- and $L_{2}(I;H^{1}(\Omega))$-type norms when $I$is the temporal and $\Omega$ the spatial domain for the PDE. We base ourmethodology for the analysis on a novel space-time reconstruction approach. Ourapproach is flexible as it works for any type of elliptic error estimator andleaves their choice of up to the user. It also allows exhibits mesh-changeestimators in a clear an concise way. We also show how our approach allows thederivation of such bounds in the $H^1(I;H^{-1}(\Omega))$ norm.

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