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RWTH Publication No: 47187 2002 IGPM225.pdf 
TITLE 
Approximation Classes for Adaptive Methods 
AUTHORS 
Peter Binev, Wolfgang Dahmen, Ronald DeVore, Pencho Petrushev 
ABSTRACT 
Adaptive Finite Element Methods (AFEM) are numerical procedures that
approximate the solution to a partial dierential 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 conver
gence [2] become available. In the latter paper it is shown that a certain AFEM
for solving Laplace's equation on a polygonal
domain
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 parti
tion P obtained from n newest vertex bisections, then the adaptively generated
solution will also use O(n) subdivisions (and 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 mea
sures of smoothness. The purpose of the present paper is to describe such
approximation classes A s by Besov smoothness.

KEYWORDS 
adaptive finite element methods, adaptive approximation, nterm approximation, degree of approximation, approximation classes, Besov spaces

PUBLICATION 
Serdica Mathematical Journal (2002), Volume: 28, Issue: 4, page 391416, ISSN: 13106600 