On uniqueness of numerical solution of boundary integral equations with 3-times monotone radial kernels

Abstract

The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function Fδ3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if δ3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than Fδ3/2. Note that there is a smart relation between δ3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln(c0r) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results.