Boundary integral equation with the generalized Neumann kernel for computing green’s function for multiply connected regions

Abstract

This research is about computing the Green’s function for both bounded and unbounded multiply connected regions by using the method of boundary integral equation. The Green’s function can be expressed in terms of an unknown function that satisfies a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystr?m method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica software and MATLAB software plots of Green’s functions for several test regions for connectivity not more than three are also presented. For bounded regions with connectivity more than three and regions with corners, the linear system is solved iteratively by using the generalized minimal residual method (GMRES) powered by fast multipole method. This method helps speed up matrix-vector product for solving large linear system and gives both fast and accurate results. MATLAB software plots of Green’s functions for several test regions are also presented.