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 the application of a smoother. 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 provide a qualitative analysis of how the error propagates throughout the algorithm, and we present a series of numerical experiments that highlight the efficiency and the computational speedup of S-AFEM.

Quasi-optimal mesh sequence construction through smoothed adaptive finite element methods / Mulita, O.; Giani, S.; Heltai, L.. - In: SIAM JOURNAL ON SCIENTIFIC COMPUTING. - ISSN 1064-8275. - 43:3(2021), pp. 2211-2241. [10.1137/19M1262097]

Quasi-optimal mesh sequence construction through smoothed adaptive finite element methods

Mulita O.
;
Giani S.;Heltai L.
2021

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 the application of a smoother. 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 provide a qualitative analysis of how the error propagates throughout the algorithm, and we present a series of numerical experiments that highlight the efficiency and the computational speedup of S-AFEM.
43
3
2211
2241
10.1137/19M1262097
https://arxiv.org/abs/1905.06924
Mulita, O.; Giani, S.; Heltai, L.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/124449
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