Numerical analysis of a non‐singular boundary integral method: Part II: The general case

Abstract

SUMMARYIn order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a ÿnite element space, a non-singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely deÿned the relation between the geometrical perturbation and the dimension of the ÿnite element space, in order to obtain a stable and convergent scheme in which there are non-singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single-layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any C ∞ domains. We prove that the properties of stability and convergence remain valid.