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Mathematics of Computation

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Adaptive Fourier-Galerkin methods

Authors: C. Canuto, R. H. Nochetto and M. Verani
Journal: Math. Comp. 83 (2014), 1645-1687
MSC (2010): Primary 65M70, 65T40
Published electronically: November 21, 2013
MathSciNet review: 3194125
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Abstract: We study the performance of adaptive Fourier-Galerkin methods in a periodic box in $\mathbb {R}^d$ with dimension $d\ge 1$. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the $hp$-FEM. We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay typical of spectral approximation. We investigate the natural sparsity class for the operator range and find that the exponential class is not preserved, thus in contrast with the algebraic class. This entails a striking different behavior of the feasible residuals that lead to practical algorithms, influencing the overall optimality. The sparsity degradation for the exponential class is partially compensated with coarsening. We present several feasible adaptive Fourier algorithms, prove their contraction properties, and examine the cardinality of the activated sets. The Galerkin approximations at the end of each iteration are quasi-optimal for both classes, but inner loops or intermediate approximations are sub-optimal for the exponential class.

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Additional Information

C. Canuto
Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
MR Author ID: 44965
ORCID: 0000-0002-8481-0312

R. H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
MR Author ID: 131850

M. Verani
Affiliation: MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
MR Author ID: 704488

Keywords: Spectral methods, adaptivity, convergence, optimal cardinality.
Received by editor(s): December 20, 2011
Received by editor(s) in revised form: December 11, 2012
Published electronically: November 21, 2013
Additional Notes: The first and the third authors were partially supported by the Italian research fund PRIN 2008 “Analisi e sviluppo di metodi numerici avanzati per EDP”
The second author was partially supported by NSF grants DMS-0807811 and DMS-1109325
Article copyright: © Copyright 2013 American Mathematical Society