Numerical Solution of the Interior Dirichlet Boundary Value Problem for the Generalized Laplace Equation by the Boundary Integral Equations Method

Abstract

In this paper we investigate a method that has been proposed by Rangogni and Occhi of approximating the interior Dirichlet boundary value problem for the generalized Laplace equation into the boundary value problem for simpler elliptic equations together with the boundary integral equations approach. Based on some assumptions the considered problem can be substituted by the Dirichlet problem for the Laplace, Klein-Gordon or Helmholtz equations. Small theoretical notes regarding the uniqueness of the solution to these boundary value problems are provided. Afterward, having fundamental solutions which are well known for each of these equations, we use the boundary integral equations method representing the solution as a single-or double-layer potential in conjunction with the quadrature method to obtain a fully discrete system of linear algebraic equations with unknown approximate values of the density at quadrature points. The well-posedness of the integral equations in appropriate spaces and the convergence analysis are also considered. Ñalculating the approximate solution of the problem for a constant-coecient equation, the approximate solution for the generalized Laplace equation is obtained as well by making one simple additional action. Finally, several numerical examples for dierent domains with dierent discretization parameters are provided in order to show the eectiveness of this approach, especially in the case of exact reduction to a constant-coecient equation.