Boundary Element Methods and Their Asymptotic Convergence

Abstract

Nowadays the most popular numerical methods for solving elliptic boundary value problems are finite differences, finite elements and, more recently, boundary element methods. The latter are numerical methods for solving integral equations (or their generalizations) on the boundary Γ of the given domain. The reduction of interior or exterior stationary boundary value problems as well as transmission problems to equivalent boundary integral equations is by no means unique, the two most popular reductions are the “direct method” and the “method of potentials”. In all these cases one needs a fundamental solution of the differential equations explicitly since it will be used in numerical computations. This restricts the boundary integral methods to cases of simple computability of a fundamental solution, i.e. essentially to differential equations with constant coefficients. The formulation on the boundary surface F reduces the dimensions of the original problem by one. For the computational treatment the boundary surface is decomposed into a finite number of segments and the boundary functions are approximated by corresponding finite elements, the boundary elements. The appropriately discretized version of the boundary integral equation then provides a finite system of linear approximate equations whose coefficient matrix, the influence matrix is fully distributed.