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
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Abstract
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.
Item Type: | Article |
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Keywords: | numerical methods, parabolic partial differential equations, finite element methods, discontinuous Galerkin, time-stepping, adaptive mesh refinement, adaptive time-step |
Schools and Departments: | School of Mathematical and Physical Sciences > Mathematics |
SWORD Depositor: | Mx Elements Account |
Depositing User: | Mx Elements Account |
Date Deposited: | 09 Feb 2021 09:15 |
Last Modified: | 14 May 2021 09:45 |
URI: | http://sro.sussex.ac.uk/id/eprint/97042 |
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