In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

In this paper, we show that we have two approaches in implementing of Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet Boundary Value Problem (BVP) for an elliptic Partial Differential Equation (PDE) with a variable coefficient. One way is by choosing the collocation points at all nodes i.e. on the boundary and interior domain. The other approach is choosing the collocation points for the interior nodes only. We present the numerical implementation of the BDIDE associated to Dirichlet BVP for an elliptic PDE with a variable coefficient by using the second approach. The BDIDE is consisting of several integrals that exhibit singularities. Generally, the integrals are evaluated by using Gauss- Legendre quadrature formula. Our numerical results show that the use of semi-analytic method gives high accuracy results. The discretized BDIDE yields a system of equations. We then apply by a direct method i.e. LU decomposition method to solve the systems of equations. In all the test domains, we present the relative errors of the solutions and the relative error for the gradient.