The use of discrete orthogonal projections in boundary element methods

Abstract

In recent papers by Sloan and Wendland Grigorie and Sloan and Grigorie Sloan and Brandts a formalismwas developed that serves many important and interesting applications in boundary element methods the commutator property for splines Based on superapproximation results this property is for example a tool of central importance in stability and convergence proofs for qualocation methods for boundary integral equations with variable coecients Another application is the transfer of superconvergence properties from constantcoecient boundary integral equations to the variable coecient case The heart of the theory is formed by the concept discrete orthogonal pro jection that arises when the Lorthogonal inner product is discretized by possibly nonstandard quadrature rules In this paper we present an overview of the theory of discrete orthogonal projections and a new set of numerical experiments that conrm the theory The main conclusion is that the presence of variable coecients of a certain smoothness does not inuence superconvergence in a negative way and that henceforth the use of superconvergencebased a posteriori error estimators in this particular case is theoretically justie