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.