The Complex Variable Boundary Element Method

In this chapter, the Complex Variable Boundary Element Method will be developed for use in approximating solutions to potential problems defined on a simply connected domain. The strategy of the method is to (1) define the problem domain Ω and boundary Γ such that there are no singularities of the state variable φ(x, y) and conjugate stream function Ψ(x, y) for z = x + iy and z ε Ω UΓ; (2) develop an approximation function which is analytic in Q and continuous on r and satisfies specified boundary conditions values on Γ; and (3) determine a relative error distribution between the approximation function and the analytic solution of the boundary value problem to be used for error analysis and subsequent modeling refinement. Steps (1) and (2) will be discussed in this chapter, with the error analysis and subsequent modeling refinement methods developed in Chapter 5.