# Approximation Classes for Adaptive Methods

Binev, Peter; Dahmen, Wolfgang; DeVore, Ronald; Petrushev, Pencho

Serdica Mathematical Journal (2002)

- Volume: 28, Issue: 4, page 391-416
- ISSN: 1310-6600

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topBinev, Peter, et al. "Approximation Classes for Adaptive Methods." Serdica Mathematical Journal 28.4 (2002): 391-416. <http://eudml.org/doc/11571>.

@article{Binev2002,

abstract = {* 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.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.},

author = {Binev, Peter, Dahmen, Wolfgang, DeVore, Ronald, Petrushev, Pencho},

journal = {Serdica Mathematical Journal},

keywords = {Adaptive Finite Element Methods; Adaptive Approximation; N-term Approximation; Degree Of Approximation; Approximation Classes; Besov Spaces},

language = {eng},

number = {4},

pages = {391-416},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Approximation Classes for Adaptive Methods},

url = {http://eudml.org/doc/11571},

volume = {28},

year = {2002},

}

TY - JOUR

AU - Binev, Peter

AU - Dahmen, Wolfgang

AU - DeVore, Ronald

AU - Petrushev, Pencho

TI - Approximation Classes for Adaptive Methods

JO - Serdica Mathematical Journal

PY - 2002

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 28

IS - 4

SP - 391

EP - 416

AB - * 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.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.

LA - eng

KW - Adaptive Finite Element Methods; Adaptive Approximation; N-term Approximation; Degree Of Approximation; Approximation Classes; Besov Spaces

UR - http://eudml.org/doc/11571

ER -

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