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S. Ferraz-Leite, D. Praetorius:
"Simple A Posteriori Error Estimators for the h-Version of the Boundary Element Method in 3D";
Talk: 4th Austrian Numerical Analysis Day, Linz; 04-24-2008 - 04-25-2008.

The h-h/2-strategy is one very basic and well-known technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. Let Φ denote the exact solution. One then considers

η_H := ||Φ_h − Φ _h/2||

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 for a mesh T_h/2 obtained from uniform refinement of T_h. We stress that η_H is always efficient - even with known efficiency constant C_eff = 1, i.e.

η_H <= ||Φ − Φ_h||.

Reliability of η_H follows immediately from the assumption

||Φ − Φ_h/2|| <= q_S ||Φ − Φ_h||

with some saturation constant q_S in (0, 1). Under this assumption, there holds

||Φ − Φ_h|| <= 1/sqrt(1-q_s²)η_H.

However, for boundary element methods, the energy norm ||·|| is non-local and thus the error estimator η_H does not provide information for a local mesh-refinement. Recent localization techniques from [Carstensen, Praetorius '06] for H^-1/2−norms allow one to replace the energy norm in the case of isotropic mesh-sequences by mesh-size weighted L2-norms. In particular, this very basic error estimation strategy can be used to steer an h-adaptive mesh-refinement. For instance, for Symm´s integral equation, the L2-norm based estimator

μ_H := ||ρ^1/2(Φ_h − Φ_h/2)||<sub>L²(Γ)

is equivalent to ηH. We thus may use μH to steer the mesh and ηH to estimate the error.

Further simplifications of the proposed error estimators ηH and μH consist in replacing Φh by some appropriate projection ΠhΦh/2, for instance, by use of the L2-projection onto the discrete space corresponding to Th. Moreover, the error estimator ηH is proven to be equivalent to the averaging estimator in [Carstensen, Praetorius '06] and the two-level estimator from [Mund, Stephan, Weisse '98].

However, the analytical results only cover the isotropic case, which in 3D doesn´t reveal the optimal order of convergence. Numerical examples using a heuristic to steer anisotropic refinements conclude the talk, suggesting that the analysis could be improved to work in the anisotropic case as well.</sub>

a posteriori error estimation, boundary element methods, symm's integral equation, averaging techniques

http://publik.tuwien.ac.at/files/pub-tm_6569.pdf