Stable multiscale bases and local error estimation for elliptic problems

authored by: Stephan Dahlke, Wolfgang Dahmen, Reinhard Hochmuth, Reinhold Schneider
Abstract: This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient a-posteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms of wavelet coefficients leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order.
External Organisation(s): RWTH Aachen University
Freie Universität Berlin (FU Berlin)
Technische Universität Darmstadt
Type: Article
Journal: Applied numerical mathematics
Volume: 23
Pages: 21-47
No. of pages: 27
ISSN: 0168-9274
Publication date: 02.1997
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.1016/S0168-9274(96)00060-8 (Access: Unknown)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{c610e7835d3f47c0a5e84a2b08ff595a,
title = "Stable multiscale bases and local error estimation for elliptic problems",
abstract = "This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient a-posteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms of wavelet coefficients leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order.",
keywords = "A-posteriori error estimators, Convergence of adaptive schemes, Elliptic operator equations, Galerkin schemes, Norm equivalences, Stable multiscale bases",
author = "Stephan Dahlke and Wolfgang Dahmen and Reinhard Hochmuth and Reinhold Schneider",
note = "Funding Information: The increasing importance of adaptive techniques in large scale computation is reflected by a vast amount of recent literature on this topic primarily in connection with finite element schemes (see, e.g., \[1-4,7,26,32,35\]). How to fit such adaptive techniques into the context of stable splittings for multilevel Schwarz type preconditioners for elliptic problems has been briefly indicated in \[30\]. On * Corresponding author. E-mail: dahmen@igpm.rwth-aachen.de. I The work of this author has been supported by Deutsche Forschungsgemeinschaft (Da 360/1-1). 2 The work of this author has been supported in part by Deutsche Forschungsgemeinschaft (Da 117/8-2). 3 The work of this author has been supported by the Graduiertenkolleg 'Analyse und Konstruktion in der Mathematik' funded by Deutsche Forschungsgemeinschaft.",
year = "1997",
month = feb,
doi = "10.1016/S0168-9274(96)00060-8",
language = "English",
volume = "23",
pages = "21--47",
journal = "Applied numerical mathematics",
issn = "0168-9274",
publisher = "Elsevier",
number = "1",
}