Numerical Solution of an Exterior Neumann Problem Using a Double Layer Potential

Introduction. Solving boundary value problems for partial differential operators by integral equation methods is not a new idea. However, the classical way to do it consists in representing the unknown solution as a potential of the type that will lead to an integral equation of the second kind. Then, Fredholm's theorems can be used. Thus, the Dirichlet problem is usually solved with the help of a double layer potential, and the Neumann problem with the use of a single layer potential. We shall have a different point of view. Our aim will be to obtain a variational formulation of the problem in order to obtain the existence and unicity of a solution and error estimates. This philosophy leads to opposite choices for the representation of the solution. Thus, J. C. Nedelec and J. Planchard, for the three-dimensional case, and M. N. Leroux for the two-dimensional case, have solved the Dirichlet problem by using a single layer potential. We propose here the solution of a Neumann problem by using a double layer potential. Let Í2 be a bounded open set of R3. Let T be the boundary of f» and f»c denote the complementary set of £7. We assume that T is sufficiently smooth, and we put the coordinates' origin in Í2. We shall write n, for the exterior normal to T, r, for the distance to the origin, [v] = ulp* i>l^xt, for the jump through T, of the function v defined in R3.