Spectral methods

Bernardi, Christine; Maday, Yvon

doi:10.1016/s1570-8659(97)80003-8

Cited by 295 publications

(227 citation statements)

References 51 publications

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“…The first one is to use the standard Hermite polynomials as the base functions (see Szegö [38], Gottlieb and Orszag [19], Canuto et al [11], Bernardi and Maday [3], and Guo [22]). In this case, ω(v) = e −v 2 .…”

Section: Some Results On Hermite Approximationmentioning

confidence: 99%

“…While the Legendre-and Chebyshev-spectral approximations for PDEs in bounded domains have achieved great success and popularity in recent years (see, e.g., [3,11,16,19]), spectral approximations for PDEs in unbounded domains have received only limited attention. Some earlier works on the convergence analysis of spectral methods in unbounded domains have been given by Funaro and Kavian [17], Guo [23] (on Hermite spectral approximations); by Mavriplis [32], Shen [37] (on Laguerre approximations); by Boyd [6], Grosch and Orszag [20] (on rational polynomial approximations).…”

Section: Introductionmentioning

confidence: 99%

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Combined Hermite spectral-finite difference method for the Fokker-Planck equation

2001

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Abstract. The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.

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Section: Some Results On Hermite Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

2001

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“…Note that in the particular case of y m,j = 0 and one-dimensional ports our port approximation is of pure Legendre-polynomial type, and we expect (based on classical results for the continuous case N γ m,j → ∞) that the eigenmode expansion will exhibit good approximation properties; the port approximation error will decay with an exponential rate with exponent linear in the number of active degrees of freedom -as long as the solution on the port is a (spatially) smooth function [4]. We would not in general expect a similar result for the "classical" non-singular Sturm-Liouville choice s m,j = 1, and we note that port reduction approaches within the CMS framework [6,18] typically consider regular rather than singular eigenproblems.…”

Section: Port Approximationmentioning

confidence: 97%

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Eftang

Patera

2013

Numerical Meth Engineering

106

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We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are i) efficient empirical port approximation spaces -the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system, and ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditionerthe error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.

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“…The rate of convergence of the error L N (u) − u is related solely to the smoothness of the function u, e.g., if u belongs to the Sobolev space H s (−1, 1), s ≥ 0, there exists a constant c (depending solely on s) such that (consult [2] for instance)…”

Section: Legendre Polynomials and Expansionsmentioning

confidence: 99%

Padé-Legendre Interpolants for Gibbs Reconstruction

2006

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We discuss the use of Padé-Legendre interpolants as an approach for the postprocessing of data contaminated by Gibbs oscillations. A fast interpolation based reconstruction is proposed and its excellent performance illustrated on several problems. Almost non-oscillatory behavior is shown without knowledge of the position of discontinuities. Then we consider the performance for computational data obtained from nontrivial tests, revealing some sensitivity to noisy data. A domain decomposition approach is proposed as a partial resolution to this and illustrated with examples.

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Spectral methods

Cited by 295 publications

References 51 publications

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Padé-Legendre Interpolants for Gibbs Reconstruction

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