EXACT SOLUTIONS Circle, annulus, sectors Rectangle, certain triangles, ellipse SEMI-ANALYTIC METHODS Boundary perturbation Domain decomposition Internal matching Variational Asymptotic Point match Galerkin Conformal map Green’s function Boundary integral

Boundary Perturbation

Domain decomposition

Internal matching

Discussions A) The best method to use depends on the problem B) With FD FE codes, why study analytic or semi-analytic methods? Analytic methods serve as a check for numerical codes which are better in more complicated situations Sharp edges, point constraints are difficult to handle for full numerical codes These methods can be applied to other elliptic pde’s such as Stokes flow.