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Journal of Scientific Computing
Published in: Journal of Scientific Computing 3/2015

01-06-2015

Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

Authors: Claudio Canuto, Valeria Simoncini, Marco Verani

Published in: Journal of Scientific Computing | Issue 3/2015

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Abstract

We analyze the theoretical properties of an adaptive Legendre–Galerkin method in the multidimensional case. After the recent investigations for Fourier–Galerkin methods in a periodic box and for Legendre–Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/\(hp\) discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in \(H^1\), based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type.
previous article A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction–Diffusion Equations on Surfaces
next article Computing Interacting Multi-fronts in One Dimensional Real Ginzburg Landau Equations

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Appendix
Available only for authorised users
Literature
1.
Adams, J.: On the expression of the product of any two legendres coefficients by means of a series of legendres coefficients. Proc. R. Soc. Lond. 27, 63–71 (1878)CrossRefMATH
2.
Baouendi, M.S., Goulaouic, C.: Régularité analytique et itérés d’opérateurs elliptiques dégénérés; applications. J. Funct. Anal. 9, 208–248 (1972)CrossRefMATHMathSciNet
3.
Benzi, M., Tůma, M.: Orderings for factorized sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21(5), 1851–1868 (electronic), 2000. Iterative methods for solving systems of algebraic equations (Copper Mountain, CO, 1998)
4.
Binev, P.: Tree approximation for \(hp\)-adaptivity. (in preparation)
5.
Binev, P.: Instance optimality for \(hp\)-type approximation. Technical Report 39, Mathematisches Forschungsinstitut Oberwolfach (2013)
6.
7.
Bürg, M., Dörfler, W.: Convergence of an adaptive \(hp\) finite element strategy in higher space-dimensions. Appl. Numer. Math. 61(11), 1132–1146 (2011)CrossRefMATHMathSciNet
8.
Canuto, C., Nochetto, R., Stevenson, R., Verani, M.: An \(hp\) adaptive finite element method: convergence and optimality properties. (in preparation)
9.
Canuto, C., Nochetto, R., Verani, M.: Contraction and optimality properties of adaptive Legendre–Galerkin methods: the 1-dimensional case. Comput. Math. Appl. 67(4), 752–770 (2014)CrossRefMathSciNet
10.
11.
Canuto, C., Simoncini, V., Verani, M.: On the decay of the inverse of matrices that are sum of Kronecker products. Linear Algebra Appl. 452, 21–39 (2014)CrossRefMATHMathSciNet
12.
Canuto, C., Verani, M.: On the numerical analysis of adaptive spectral/hp methods for elliptic problems. In: Brezzi, F. et al. (eds.) Analysis and Numerics of Partial Differential Equations, pp. 165–192. Springer INdAM series (2013)
13.
Cascon, J.M., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)CrossRefMATHMathSciNet
14.
Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comput. 70, 27–75 (1998)CrossRefMathSciNet
15.
Cohen, A., DeVore, R., Nochetto, R.H.: Convergence rates of AFEM with \(H^{-1}\) data. Found. Comput. Math. 12(5), 671–718 (2012)CrossRefMATHMathSciNet
16.
17.
Dörfler, W., Heuveline, V.: Convergence of an adaptive \(hp\) finite element strategy in one space dimension. Appl. Numer. Math. 57(10), 1108–1124 (2007)CrossRefMATHMathSciNet
18.
Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Clarendon Press, Oxford (1989)MATH
19.
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD (1996)
20.
Gui, W., Babuška, I.: The \(h,\;p\) and \(h\)-\(p\) versions of the finite element method in 1 dimension. III. The adaptive \(h\)-\(p\) version. Numer. Math. 49(6), 659–683 (1986)CrossRefMATHMathSciNet
21.
Hall, P., Jin, J.: Innovated higher criticism for detecting sparse signals in correlated noise. Ann. Stat. 38(3), 1686–1732 (2010)CrossRefMATHMathSciNet
22.
Jaffard, S.: Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. Annales de l’I.H.P. 5, 461–476 (1990)
23.
24.
Maitre, J.-F., Pourquier, O.: Condition number and diagonal preconditioning: comparison of the \(p\)-version and the spectral element methods. Numer. Math. 74(1), 69–84 (1996)CrossRefMATHMathSciNet
25.
Mitchell, W.F., McClain, M.A.: A survey of \(hp\)-adaptive strategies for elliptic partial differential equations. In: Recent Advances in Computational and Applied Mathematics, pp. 227–258. Springer, Dordrecht (2011)
26.
Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000). (electronic)CrossRefMATHMathSciNet
27.
Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)
28.
Schmidt, A., Siebert, K.G.: A posteriori estimators for the \(h\)-\(p\) version of the finite element method in 1D. Appl. Numer. Math. 35(1), 43–66 (2000)CrossRefMATHMathSciNet
29.
30.
Stevenson, R.: Adaptive wavelet methods for solving operator equations: an overview. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 543–597. Springer, Berlin (2009)
Metadata
Title
Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case
Authors
Claudio Canuto
Valeria Simoncini
Marco Verani
Publication date
01-06-2015
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 3/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-014-9912-3

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