Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods

Mulita, Ornela; Giani, Stefano; Heltai, L.

doi:10.1137/19m1262097

Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods

Mulita, Ornela; Giani, Stefano; Heltai, L.

Authors

Ornela Mulita

Dr Stefano Giani stefano.giani@durham.ac.uk
Associate Professor

L. Heltai

Abstract

We propose a new algorithm for Adaptive Finite Element Methods (AFEMs) based on smoothing iterations (S-AFEM), for linear, second-order, elliptic partial differential equations (PDEs). The algorithm is inspired by the ascending phase of the V-cycle multigrid method: we replace accurate algebraic solutions in intermediate cycles of the classical AFEM with the application of a prolongation step, followed by a fixed number of few smoothing steps. Even though these intermediate solutions are far from the exact algebraic solutions, their a-posteriori error estimation produces a refinement pattern that is substantially equivalent to the one that would be generated by classical AFEM, at a considerable fraction of the computational cost. We quantify rigorously how the error propagates throughout the algorithm, and we provide a connection with classical a posteriori error analysis. A series of numerical experiments highlights the efficiency and the computational speedup of S-AFEM.

Citation

Mulita, O., Giani, S., & Heltai, L. (2021). Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods. SIAM Journal on Scientific Computing, 43(3), A2211-A2241. https://doi.org/10.1137/19m1262097

Journal Article Type	Article
Acceptance Date	Mar 30, 2021
Online Publication Date	Jun 17, 2021
Publication Date	2021
Deposit Date	Mar 31, 2021
Publicly Available Date	Mar 31, 2021
Journal	SIAM Journal on Scientific Computing
Print ISSN	1064-8275
Electronic ISSN	1095-7197
Publisher	Society for Industrial and Applied Mathematics
Peer Reviewed	Peer Reviewed
Volume	43
Issue	3
Pages	A2211-A2241
DOI	https://doi.org/10.1137/19m1262097

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Copyright Statement
First Published in SIAM journal on scientific computing in 43:3, 2021, published by the Society for Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.