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Talks and Poster Presentations (without Proceedings-Entry):

M. Aurada, S. Ferraz-Leite, D. Praetorius:
"Convergence of adaptive BEM";
Talk: 7th Söllerhaus Workshop on Fast Boundary Element Methods in Industrial Applications, Hirschegg; 10-15-2009 - 10-18-2009.



English abstract:
A posteriori error estimators and adaptive mesh-refinement have themselves proven to be an important tool for scientific computing. For error control in finite element methods (FEM), there is a broad variety of a posteriori error estimators available, and convergence as well as optimality of adaptive FEM is well-studied in the literature. This is in sharp contrast to the boundary element method (BEM). Although a posteriori error estimators and adaptive algorithms are also successfully applied to boundary element schemes, even convergence of adaptive BEM is hardly understood mathematically. In our contribution, we present and discuss recent mathematical results which give first positive answers for adaptive BEM.

As BEM model problem for our talk serves the weakly-singular integral equation

Vu=f

associated with the Laplace operator in 2D and 3D and stated in the energy space H−1/2(Γ). We use the lowest-order Galerkin method with piecewise constant ansatz and test functions and consider standard adaptive algorithms of the type

Solve --> Estimate --> Mark --> Refine

It is a simple consequence of functional analysis that the sequence uk of Galerkin solutions generated by this algorithm, tends to some limit uinfinity in H−1/2(Γ). It is, however, a priori unknown whether uinfinity coincides with the unique exact solution u in H−1/2(Γ) of the integral equation.

For a posteriori error estimation, we use certain (h − h/2)-type error estimators μk, and element marking is done by the criterion introduced by Dörfler. We then treat the convergence

limk-->infinityuk=u

as well as

limk-->infinityμk=0

for both, isotropic and anisotropic mesh-refinement.

Keywords:
boundary element methods, convergence, adaptive algorithm, error estimator, anisotropic mesh-refinement