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

Mulita, Ornela and Giani, Stefano and Heltai, L. (2021) 'Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods.', SIAM journal on scientific computing., 43 (3). A2211-A2241.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1137/19M1262097
Publisher 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.
Date accepted:30 March 2021
Date deposited:31 March 2021
Date of first online publication:17 June 2021
Date first made open access:31 March 2021

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