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

TitleAdaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods
Publication TypeJournal Article
Year of Publication2021
AuthorsCangiani, A, Georgoulis, EH, Sutton, OJ
JournalMathematical Models and Methods in Applied Sciences
Volume31
Pagination711-751
Abstract

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.

URLhttps://doi.org/10.1142/S0218202521500172
DOI10.1142/S0218202521500172

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