Local a posteriori error estimates for boundary element methods

We consider an integral equation Au = f with a strongly elliptic integral operator A and given right-hand side f. For solving that equation we use the spline Galerkin method. We begin with some localized error estimates which are based on a localization of the error equation and the commutator property for pseudodifferential operators. One obtains an error estimate on fixed parts of the boundary surface with remainder terms which are asymptotically of smaller order with respect to the meshsize than the error itself. To prove mesh-dependent localized error estimates, the main difficulty is the generalization of the commutator property of pseudodifferential operators for non-smooth truncation functions. We obtain a mesh-dependent localized error estimate with an explicit relation between the smoothness of the truncation function which is related to the size of the local supports and the size of the remainder terms.