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

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Version 2 2023-06-12, 09:43
Version 1 2023-06-09, 23:01
journal contribution
posted on 2023-06-12, 09:43 authored by Emmanuil H Georgoulis, Omar LakkisOmar Lakkis, Thomas P Wihler
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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Numerische Mathematik

ISSN

0029-599X

Publisher

Springer

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2021-02-09

First Open Access (FOA) Date

2021-05-14

First Compliant Deposit (FCD) Date

2021-02-09

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