Convergence rates for adaptive finite element methods

Anders Szepessy

Abstract

There are numerous adaptive methods for partial differential equations, but the theoretical understanding of convergence rates for adaptive algorithms is not as well understood, with a good exception for the wavelet methods by Cohen, Dahmen and DeVore.

I will present a rigorous convergence result for a general adaptive finite element algorithm based on successive subdivision of the mesh: the algorithm decreases the maximal error indicator with a constant factor until it stops; the algorithm stops with the optimal number of elements, up to a problem independent factor; and the global error is asymptically bounded by the tolerance parameter, up to a problem independent factor, as the tolerance tends to zero.

The talk will answer how to measure the convergence rate ( i.e. what is the optimal number of elements?), why the convergence of the error density for the error indicators is subtle and why this convergence is essential.