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S. Ferraz-Leite, C. Ortner, D. Praetorius:
"Convergence of h-h/2 based adaptive Galerkin schemes";
Talk: 5th Austrian Numerical Analysis Day, Innsbruck; 05-07-2009 - 05-08-2009.

Let H be a real Hilbert space with scalar product <.,.> and
associated energy norm ||.||. Given a right-hand side F in H^*,
we aim at a numerical approximation of the unique solution u in H of

<u,v> = F(v) for all v in H.

To that end, let X_l be a finite-dimensional subspace of H,
which in usual applications is based on a triangulation. Let u_l in X_l be the corresponding Galerkin
solution, obtained by solving the finite-dimensional linear system

<u_l,v_l> = F(v_l) for all v_l in X_l.

Throughout, the index l = {0, 1, 2, ...} denotes the
step of an adaptive algorithm.
We say that such an algorithm is convergent if, and
only if, u_l tends to u as l tends to infinity.

We prove that the saturation assumption yields convergence of
an adaptive algorithm based on the h-h/2-error estimator.
Besides the adaptive finite element method (AFEM), our argument
applies to boundary integral formulations and yields a first
convergence proof for the adaptive boundary element method (ABEM).

adaptive algorithm, BEM, FEM, convergence, Galerkin method

http://publik.tuwien.ac.at/files/PubDat_175483.pdf