Simple A Posteriori Error Estimators For The H-version Of The Boundary Element Method

摘要

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 η := ‖φ_(h/2) - φ_h ‖ to estimate the error ‖φ -φ_h‖, where φ_h is a Galerkin solution with respect to a mesh T_h and φ_(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 η 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~(-1/2) (Γ). 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~(1/2) (Γ) and H_0~(1/2)(Γ), respectively, or a transmission problem.