Publications in Scientific Journals:

C. Carstensen, D. Praetorius:
"A posteriori error control in adaptive qualocation boundary element analysis for a logarithmic-kernel integral equation of the first kind";
SIAM Journal on Scientific Computing, 25 (2004), 1; 259 - 283.

English abstract:
Sobolev-Slobodeckij norms on small overlapping domains of two
neighbouring elements serve as a~posteriori error estimators and
mesh-refining indicators in adaptive boundary element methods.
The paper is concerned with two error estimators $\eta_F$ and $\mu_F$.
The first variant $\eta_F$ is efficient and the second $\mu_F$ is reliable,
that is, up to multiplicative constants and numerical quadrature
errors, they are lower or upper error bounds.
Faermann recently established reliability and efficiency
of $\eta_F$ for the Galerkin boundary element method and
considered $\mu_F$.

This work approaches the two estimators $\eta_F\le\mu_F$ for
the Galerkin, qualocation, and collocation boundary element methods
for a single layer operator integral equation of the first kind.
Upper and lower bounds are established theoretically and validated
numerically. Numerical experiments support the estimators' accuracies
and the efficiencies of proposed adaptive mesh-refining algorithms
even in energy norms. For qualocation and collocation schemes,
difficulties for $\alpha\le 1/2$ are caused by the lack of a
Sobolev embedding $H^\alpha(\Gamma)\hookrightarrow\CC(\Gamma)$.
Hence, for the latter schemes, equivalence of error and estimators in
$H^{\alpha-1}(\Gamma)$ can be proven only for $\alpha>1/2$. Numerical
evidence conjectures equivalence for $\alpha=1/2$ as well.

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