Stability, convergence and order of the extrapolations of the residual smoothing scheme in energy norm
type: Item
instance of: scholarly article
title: Stability, convergence and order of the extrapolations of the residual smoothing scheme in energy norm
zbMATH Open document ID: 1230.65103
DOI: 10.1142/S1793744211000436
author: M. Ribot
author: Michelle Schatzman
published in: Confluentes Mathematici
publication date: 2011-11-02T00:00:00Z
review text: The authors consider an abstract differential equation in a Hilbert space having in view application of the method of lines to parabolic equations discretized by spectral or finite element methods. The residual smoothing scheme consists in preconditioning the discrete time derivative of the explicit Euler method by a suitable operator. This, like the elliptic operator, is assumed to be self-adjoint and non-negative, further, it contains a free (stability) parameter. After introducing order relations for such operators and proving that the one-step operator has energy norm not exceeding 1, they investigate properties of Richardson extrapolation. The resulting operator is shown to be stable in the energy norm for sufficiently large values of the stability parameter and to assure convergence with any fixed order \(k\) in time. Comparing with results of the alternating directions implicit (ADI) time method we can say that about fifty years ago cheaper ways of preconditioning parabolic equations were found and shown to converge, see Yanenko (1959), Bramble, D'yakonov and Samarskij -- where the Russian papers all were translated to English [e.g., \textit{N. N. Yanenko}, The method of fractional steps. The solution of problems of mathematical physics in several variables. Berlin-Heidelberg-New York: Springer Verlag (1971; Zbl 0209.47103)]. True, the operators were imagined to arise from finite difference methods, and higher order was reached for \(d\)-dimensional parallelepipedons only. However, the stability theory of Samarskij was assuming essentially just the same properties like the present paper: self-adjointness, spectral equivalence of elliptic operator and preconditioner, and resulted convergence in energy norm (a German translation appeared in 1984 of his work from the seventies, see \textit{A. A. Samarskij} [Theorie der Differenzenverfahren. Mathematik und ihre Anwendungen in Physik und Technik, Bd. 40. Leipzig: Akademische Verlagsgesellschaft Geest \& Portig K.-…
reviewed by: Gisbert Stoyan
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65F08
zbMATH DE Number: 5966734
zbMATH Keywords: residual smoothing scheme
zbMATH Keywords: parabolic equations
zbMATH Keywords: convergence
zbMATH Keywords: energy norm
zbMATH Keywords: spectral method
zbMATH Keywords: abstract differential equation
zbMATH Keywords: Hilbert space
zbMATH Keywords: method of lines
zbMATH Keywords: finite element methods
zbMATH Keywords: preconditioning
zbMATH Keywords: Euler method
zbMATH Keywords: Richardson extrapolation
zbMATH Keywords: stability
zbMATH Keywords: finite difference methods
MaRDI profile type: Publication
full work available at URL: s1793744211000436
OpenAlex ID: W2093453151
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description: scientific article; zbMATH DE number 5966734