

Rob Stevenson : An optimal adaptive finite element method
We present an adaptive finite element method for solving second order
elliptic equations which is (quasi)optimal in the following sense: If
the solution is such that for some s>0, the errors in energy norm of the
best continuous piecewise linear approximations subordinate to any
partition with N triangles are O(N^{s}), then given a tol>0, the
adaptive method produces an approximation with an error less than
tol subordinate to a partition with O(tol^{1/s}) triangles, taking only O(tol^{1/s})operations.
Our method is based on ideas from [Binev, Dahmen, DeVore '02], in which
a coarsening routine was added to the method from
[Morin, Nochetto, Siebert '00].
Differences are that we employ nonconforming partitions, our coarsening
routine is based on a transformation to a wavelet basis, all our results
are valid uniformly in the size of possible jumps of the diffusion coefficients,
and that we allow more general righthand sides. All tolerances in our
adaptive method depend on a posteriori estimate of the current error
instead an a priori one, which can be expected to give quantitative advantages.
