Simple a posteriori error estimators for the h-version of the boundary element method
The h-h/2-strategy is one well-known technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. One considers eta := ||phi(h/2) - phi(h)|| to estimate the error ||phi - phi(h)||, where phi(h) is a Galerkin solution with respect to a mesh T(h), and phi(h/2) is a Galerkin solution with respect to the mesh T(h/2) obtained from a uniform refinement of T(h). This error estimator is always efficient and observed to be also reliable in practice. However, for boundary element methods, the energy norm is non-local and thus the error estimator 77 does not provide information for a local mesh-refinement. We consider Symm's integral equation of the first kind, where the energy space is the negative-order Sobolev space (H) over tilde (-1/2) (Gamma). Recent localization techniques allow to replace the energy norm in this case by some weighted L(2)-norm. Then, this very basic error estimation strategy is also applicable to steer an h-adaptive algorithm. Numerical experiments in 2D and 3D show that the proposed method works well in practice. A short conclusion is concerned with other integral equations, e.g., the hypersingular case with energy space (H) over tilde (1/2) (Gamma) and H(0)(1/2) (Gamma), respectively, or a transmission problem.
a posteriori error estimate boundary element method symm's integral equation galerkin method adaptive algorithm