Boundary Integral Equations - The Boundary Element Method with Programming - page 92

5
Boundary Integral Equations

There is nothing more practical
than a good theory

I. Kant

5.1 INTRODUCTION

As explained previously, the basic idea of the boundary element method comes from
Trefftz 1 , who suggested that in contrast to the method of Ritz, only functions satisfying
the differential equations exactly should be used to approximate the solution inside the
domain. If we use these functions it means, of course, that we only need to approximate
the actual boundary conditions. This approach, therefore, has some considerable
advantages:

x The solutions obtained inside the domain satisfy the differential equations exactly,
approximations (or errors) only occur due to the fact that boundary conditions are
only satisfied approximately.
x Since functions are defined globally, there is no need to subdivide the domain into
elements.
x The solutions also satisfy conditions at infinity, therefore, there is no problem dealing
with infinite domains, where the FEM has to use mesh truncation or approximate
infinite elements.

The disadvantage is that we need solutions of differential equations to be as simple as
possible, if we want to reduce computation time. The most suitable solutions are ones