Approximation Classes for Adaptive Methods

* 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.