225 | IGPM225.pdf December 2002 |
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, n-term approximation, degree of approximation, approximation classes, Besov spaces |
PUBLICATION | Serdica Math. J. 28 (2002), 1001--1026 |