A Posteriori Error Control in Adaptive Qualocation Boundary Element Analysis for a Logarithmic-Kernel Integral Equation of the First Kind

Abstract

Sobolev-Slobodeckij norms on small overlapping domains of two neighboring elements serve as a posteriori error estimators and mesh-refining indicators in adaptive boundary element methods. This 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 $H^\alpha(\Gamma)\hookrightarrow{\mathcal{C}} (\Gamma)$. 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.