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Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

journal contribution
posted on 26.08.2021, 07:53 by A Cangiani, EH Georgoulis, OJ Sutton
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L∞(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.

History

Citation

Mathematical Models and Methods in Applied Sciences, Vol. 31, No. 04, pp. 711-751 (2021)

Author affiliation

School of Mathematics & Actuarial Science

Version

AM (Accepted Manuscript)

Published in

Mathematical Models and Methods in Applied Sciences

Volume

31

Issue

4

Pagination

711 - 751

Publisher

World Scientific Publishing

issn

0218-2025

eissn

1793-6314

Acceptance date

30/12/2020

Copyright date

2021

Available date

17/04/2022

Notes

31 pages, 27 figures

Language

eng

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