Numerical analysis of a non-singular boundary integral method: Part I. The circular case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non-singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright 2001 John Wiley & Sons, Ltd.