Galerkin Boundary Element Method For Harmonic Equation With Open Boundary

The harmonic equation,or Laplace equation is a typical and simple elliptic partial differential equation. The mathematical model of many problems of mechanics and physics are reduced to boundary problems of Laplace equation. Such as: the balance of elastic membrane, the heat conduction of stationary state, the incompressible potential flow, the problem of electrostatic field and magnetostatics field. These problems have different physical nature, but can be reduced to the same mathematical expression. In this paper, we study the numerical solutions of these problems using Galerkin boundary element, especially for the problem with open boundary such as the problems exterior to an open segment or an open curve in the plane. Since the equivalent boundary integral equation for two dimensional Laplace equation has constraint condition. Lagrange multiplicator method is introduced in the numerical computation to release the constraint. Galerkin method based on the variation principle is used to solve differential and integral equations. In this paper, the boundary problem of Laplace equation is changed into the variational equation which is equivalent to the boundary integral equation. Using linear element, it is solved by Galerkin boundary element method. In computation of stiffness matrix, the calculation of double singular integration is needed. The exactly integral formula is used in the inner integral expression, and the numerical integral formula is used in the outer integral expression. We extend this method to the problem of a domain which boundary is an open arc or an open segment. In this paper, we simulate the singularity of solution in extreme point of open arc or open segment by singular element. With Fortran Power Station 4.0, we make Galerkin boundary element method program for solving Laplace equation on region which boundary is a closed curve or an open arc, and the numerical experiments also prove this method is reliable. Last we test the error of Galerkin boundary element by numerical experimentation.