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

Abstract

SUMMARYIn 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 ÿnite element space, a non-singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely deÿne the relation between the geometrical perturbation and the dimension of the ÿnite 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 deÿned 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.

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

“…We can remark that in the circular case, i.e. when is a circle, then Theorem 2.1 is exactly Theorem 2.1 of Reference [1]. The object here is to generalize the proof for any C ∞ closed curve.…”

“…Remark that in (7) the symbol 'integral' takes the meaning of duality : ; : −1=2; 1=2 . As we said in the introduction of our previous paper [1], problem (7) is ill-posed because the continuous bilinear form a n :…”

“…In Reference [1] we have presented a complete history of the method: it was introduced in its ÿrst form by Kupradze [3] and studied by Christiansen (cf. References [4; 5]).…”

“…for instance References [6][7][8]). Our contribution in this work and in Reference [1] is to mathematically prove a stability property of the method and to establish results of convergence. Moreover, we give the rules for the choice of a quadrature formula when we want to numerically compute n .…”

“…Moreover, we give the rules for the choice of a quadrature formula when we want to numerically compute n . We have also explained in Reference [1] how the proposed scheme can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the single-layer potential. From this point of view, the proposed method is related to a class that has been studied for many years, from the early papers [9; 10] up to recent papers (Reference [11] for instance).…”