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

Georgoulis, Emmanuil H.; Lakkis, Omar; Wihler, Thomas P. (2021). A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems. Numerische Mathematik, 148(2), pp. 363-386. Springer-Verlag 10.1007/s00211-021-01187-7

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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 L_∞(I; L_2(Ω))- and L_2(I; H1(Ω))-type norms when I is the temporal and Ω 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 H^1(I; H^{−1}(Ω)) norm.

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