Boundary Integral Equations

In Chapter 10, we examined finite element methods for the numerical solution of Laplace's equation. In this chapter, we propose an alternative approach. We introduce the idea of reformulating Laplace's equation as a boundary integral equation (BIE), and then we consider the numerical solution of Laplace's equation by numerically solving its reformulation as a BIE. Some of the most important boundary value problems for elliptic partial differential equations have been studied and solved numerically by this means; and depending on the requirements of the problem, the use of BIE reformulations may be the most efficient means of solving these problems. Examples of other equations solved by use of BIE reformulations are the Helmholtz equation (Δu + λu = 0) and the biharmonic equation (Δ²u = 0). We consider here the use of boundary integral equations in solving only planar problems for Laplace's equation. For the domain D for the equation, we restrict it or its complement to be a simply-connected set with a smooth boundary S. Most of the results and methods given here will generalize to other equations (e.g. Helmholtz's equation).