ON NUMERICAL CUBATURES OF SINGULAR SURFACE INTEGRALS IN BOUNDARY ELEMENT METHODS

We present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in R3. The class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three-dimensional domains by the boundary integral equation method. The possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point. This rules out weighted quadrature approaches with a fixed singular weight. For weakly singular integrals it is shown that Duffy's triangular coordinates lead always to a removal of the kernel singularity. Also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero. These are based on the Rabinowitz-Richter estimates in connection with an asymptotic estimate of domains of analyticity in C2. It is further shown that the modified extrapolation approach due to Lyness is in the weakly singular case always applicable. Corresponding error and asymptotic work estimates are presented. For the weakly singular as well as for Cauchy and hypersingular integrals which e.g. arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates. These rules are hierarchically constructed from "finite part" integration formulas in radial and Gaussian formulas in angular direction. Again, we show how the Rabinowitz-Richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size.