Energy norm based a posteriori error estimation for boundary element methods in two dimensions

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. We analyze the mathematical relation between the h-h/2-error estimator from [S. Ferraz-Leite, D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008) 135-162], the two-level error estimator from [S. Funken, Schnelle Losungsverfahren fur FEM-BEM Kopplungsgleichungen, Ph.D. thesis, University of Hannover, 1996 (in German); P. Mund, E. Stephan, J. Weisse, Two-level methods for the single layer potential in R^3, Computing 60 (1998) 243-266], and the averaging error estimator from [C. Carstensen, D. Praetorius, Averaging techniques for the effective numerical solution of Symm's integral equation of the first kind, SIAM J. Sci. Comput. 27 (2006) 1226-1260]. We essentially show that all of these are equivalent, and we extend the analysis of [S. Funken, Schnelle Losungsverfahren fur FEM-BEM Kopplungsgleichungen, Ph.D. thesis, University of Hannover, 1996 (in German); P. Mund, E. Stephan, J. Weisse, Two-level methods for the single layer potential in R^3, Computing 60 (1998) 243-266] to cover adaptive mesh-refinement. Therefore, all error estimators give lower bounds for the Galerkin error, whereas upper bounds depend crucially on the saturation assumption. As model examples, we consider first-kind integral equations in 2D with weakly singular integral kernel.

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