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

In order to numerically solve the interior and the exterior Dirichlet probl ems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non-singular integral operator. Thi s operator is a geometrical perturbation of the Steklov operator, and we pr ecisely define the relation between the geometrical perturbation and the di mension of the finite element space, in order to obtain a stable and conver gent scheme. Furthermore, this numerical scheme does not give rise to any s ingular integral. The scheme can also be considered as a special quadrature formula method fo r the standard piecewise linear Galerkin approximation of the weakly singul ar single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove st ability and we give error estimates of our numerical scheme when the Laplac e problem is set on a disk. We will extend our results to any domains by us ing compact perturbation arguments, in a second paper. Copyright 2001 John Wiley & Sons, Ltd.