Boundary Element Methods

Boundary element methods are recent developments in computational mathematics for solving boundary value problems in various branches of science and technology. These methods evolved from integral equation methods which are known as boundary integral equation methods (BIEM’s). There are two types of BIEM’s which, though seemingly alike, had totally different approaches of formulation. One of them is the so called ‘indirect’ method, which relies on the physical aspect of the problem and involves the transformation of the boundary surface (or curve) by a surface (or curve) of sources/sinks of adjustable strengths. The other, known as the ‘direct’ method, is based on finding the Green’s function solutions of partial differential equations. The methodology uses the fact that if the Green’s function of a given equation, together with the prescribed boundary conditions on a geometrically well-defined boundary, is known, then the solution of such a boundary value problem is also known in the form of an integral equation and can be numerically computed. Once it was found that the use of the Green’s function in the free space, i.e., the fundamental solution, would reduce the dimension of the problem by unity in the homogeneous case, whereby the volume integrals reduce to surface integrals, the surface integrals to line integrals, and the governing differential equations to integral equations,these methods thereafter became very popular during the period from about 1960 through 1975. The work of Jawson (1963), Symm (1963), and Cruse and Rizzo (1968) are the forerunners of what was to be known as the boundary element method (BEM). This method is based on integral equation formulation of boundary value problems, requires discretization of only the boundary (surface or curve) and not the interior of the region under consideration, and is suitable for problems with complicated boundaries, unbounded regions, and free surfaces.