Optimal adaptive finite element methods
Rob Stevenson (Utrecht University)

Nowadays, adaptive finite element methods are an indispensable tool
for solving complicated (systems of) partial differential
equations. Often, in very localized regions, the solution of such
equations is much less smooth than elsewhere, and therefore
approximations with respect to quasi-uniform meshes are not
efficient. The theory of nonlinear approximation, together with Besov
regularity theorems show that potentially adaptive methods can break
complexity barriers that exist for their non-adaptive counterparts. On
the other hand, although in use for more than 25 years, it was not
before '98 that, in more than one dimension, even convergence of
adaptive methods was demonstrated. Recently, however, quite some
progress was made in the sense that it has been shown that an adaptive
method, that is in use in practice, realizes the best possible rates,
and so indeed breaks the aforementioned complexity barrier. In this
talk I'll review these developments, and discuss some open problems
and possible future developments.