Self-regular boundary integral equation formulations for Laplace's equation in 2-D

The purpose of this work is to demonstrate the application of the self-regu lar formulation strategy using Green's identity (potential-BIE) and its gra dient form (flux-BIE) for Laplace's equation. Self-regular formulations lea d to highly effective BEM algorithms that utilize standard conforming bound ary elements and low-order Gaussian integrations. Both formulations are dis cussed and implemented for two-dimensional potential problems, and numerica l results are presented. Potential results show that the use of quartic int erpolations is required for the flux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, flux erro r results in the potential-BIE implementation can be dominated by the numer ical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these flux results does not improve beyond a certain level when using standard quadrature together with a special transformatio n, but when an alternative logarithmic quadrature scheme is used these erro rs are shown to reduce abruptly, and the flux results converge monotonicall y to the exact answer. In the flux-BIE implementation, where all integrals are regularized, flux results accuracy improves systematically, even with s ome oscillations, when refining the mesh or increasing the order of the int erpolating function. The flux-BIE approach presents a great numerical sensi tivity to the mesh generation scheme and refinement. Accurate results for t he potential and the flux were obtained for coarse-graded meshes in which t he rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes wer e not found in the elasticity problem for which self-regular formulations h ave also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular p otential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular p otential-BIE is compared with the standard (CPV) formulation, showing the e quivalence between these formulations. The self-regular BIE formulations an d computational algorithms are established as robust alternatives to singul ar BIE formulations for potential problems. Copyright (C) 2001 John Wiley & Sons, Ltd.