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Talks and Poster Presentations (without Proceedings-Entry):

M. Aurada, S. Ferraz-Leite, D. Praetorius:
"Convergence of Adaptive BEM";
Talk: MAFELAP 2009 - The Mathematics of Finite Elements and Applications, Uxbridge (invited); 06-09-2009 - 06-12-2009.



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 associated with the Laplace operator in 2D and 3D and stated
in the energy space $\widetilde H^{-1/2}(\Gamma)$. 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
$u_\ell$ of Galerkin solutions generated by this algorithm, tends
to some limit $u_\infty\in\widetilde H^{-1/2}(\Gamma)$. It is,
however, a priori unknown whether $u_\infty$ coincides with the
unique exact solution $u\in\widetilde H^{-1/2}(\Gamma)$ of the
integral equation.

For a posteriori error estimation, we use certain $(h-h/2)$-type
error estimators $\mu_\ell$, and element marking is done by the
$\ell_2$-criterion introduced by Doerfler. We then treat the
convergence for both, isotropic and anisotropic mesh-refinement.

Keywords:
boundary element method, a posteriori error estimate, adaptive mesh-refinement, convergence of adaptive algorithm

Created from the Publication Database of the Vienna University of Technology.