Galerkin Boundary Element Method For Direct Boundary Integral Equation For Dirichlet Problem Of 2-D Laplace Equation

The Laplace equation is a typical and simplest elliptic partial differential equation with wide applications. When using the boundary element method to solve the boundary-value problems of Laplace equation, different boundary reduction approaches will lead to different boundary integral equation,and there exist several methods to solve the boundary integral equation. In this paper the direct boundary integral equation of two-dimensional Laplace equation for Dirichlet problem is considered, which is deduced by Green's formula and the fundamental solution, and is a Fredholm integral equation of the first kind. The general Fredholm integral equation of the first kind with logarithmic kernel in 2 dimensions does not always have unique solution, especially for the unbounded problem, it depends on the asymptotic behaviour at infinity. The solvability can not be sidestepped, for it relates the problem of equivalence of boundary reduction. In fact, for interior problem, solution of the direct boundary integral equation always exists. However, for exterior problem, in order to ensure the uniqueness of the solution, the behavior of the solution at infinity must be specified, thus the modification for the expression of the solution and the constraint for the solution of boundary integral equation are needed. The most-used numerical method for solving direct boundary integral equation is collocation method, which is simple and faster. Seldom have been used the Galerkin scheme in this case. In this paper, the Galerkin boundary element method is used to solve the direct boundary integral equation in order to compare the efficiency and precision of the two methods, and the convergence results are more easily found for it. Galerkin scheme, is a variational approach, shifts the integral equation to a variational equation in which double integrations shall be carried out. The paper presents the analytical formula to calculate the inner integration and the Gaussian quadrature is used for the outer integration. For exterior Dirichlet problem, the condition, that the integration of the solution of the direct first kind integral on the boundary should be zero, does not necessarily satisfied, which turns to be a constraint condition attached to the integral equation. In this paper, a Lagrange multiplier is introduced to replace the constraint, an extended vatiational equation which specifies the behavior at infinity is solved by boundary elements, Constant element and linear element are used in the computing program of this paper, which is written by Fortran90. The numerical experiments tested the relation of error with the numbers of elements, which demonstrated that the method in this paper is reliable and effective. We can conclude that Galerkin scheme is better than collocation method in computing precision.