Details
| Original language | English |
|---|---|
| Pages (from-to) | 21-47 |
| Number of pages | 27 |
| Journal | Applied numerical mathematics |
| Volume | 23 |
| Issue number | 1 |
| Publication status | Published - Feb 1997 |
| Externally published | Yes |
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
ASJC Scopus subject areas
- Mathematics(all)
- Numerical Analysis
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Applied numerical mathematics, Vol. 23, No. 1, 02.1997, p. 21-47.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Stable multiscale bases and local error estimation for elliptic problems
AU - Dahlke, Stephan
AU - Dahmen, Wolfgang
AU - Hochmuth, Reinhard
AU - Schneider, Reinhold
N1 - 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.
PY - 1997/2
Y1 - 1997/2
N2 - 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.
AB - 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.
KW - A-posteriori error estimators
KW - Convergence of adaptive schemes
KW - Elliptic operator equations
KW - Galerkin schemes
KW - Norm equivalences
KW - Stable multiscale bases
UR - http://www.scopus.com/inward/record.url?scp=0031069391&partnerID=8YFLogxK
U2 - 10.1016/S0168-9274(96)00060-8
DO - 10.1016/S0168-9274(96)00060-8
M3 - Article
AN - SCOPUS:0031069391
VL - 23
SP - 21
EP - 47
JO - Applied numerical mathematics
JF - Applied numerical mathematics
SN - 0168-9274
IS - 1
ER -