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Title: Approximation Classes for Adaptive Methods
Authors: Binev, Peter
Dahmen, Wolfgang
DeVore, Ronald
Petrushev, Pencho
Keywords: Adaptive Finite Element Methods
Adaptive Approximation
N-term Approximation
Degree Of Approximation
Approximation Classes
Besov Spaces
Issue Date: 2002
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 28, No 4, (2002), 391p-416p
Abstract: Adaptive Finite Element Methods (AFEM) are numerical procedures that approximate the solution to a partial differential equation (PDE) by piecewise polynomials on adaptively generated triangulations. Only recently has any analysis of the convergence of these methods [10, 13] or their rates of convergence [2] become available. In the latter paper it is shown that a certain AFEM for solving Laplace’s equation on a polygonal domain Ω ⊂ R^2 based on newest vertex bisection has an optimal rate of convergence in the following sense. If, for some s > 0 and for each n = 1, 2, . . ., the solution u can be approximated in the energy norm to order O(n^(−s )) by piecewise linear functions on a partition P obtained from n newest vertex bisections, then the adaptively generated solution will also use O(n) subdivisions (and floating point computations) and have the same rate of convergence. The question arises whether the class of functions A^s with this approximation rate can be described by classical measures of smoothness. The purpose of the present paper is to describe such approximation classes A^s by Besov smoothness.
Description: * This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation grants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401, the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan- der von Humboldt Foundation.
ISSN: 1310-6600
Appears in Collections:Volume 28 Number 4

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