Fast Numerical Solution Of Exterior Helmholtz Problems With Radiation Boundary Condition By Imbedding

The development of efficient solution algorithms for Poisson's equation on domains allowing for separation of variables prompted research towards extending these algorithms to domains of general shape. The resulting numerical techniques are known as imbedding methods or capacitance matrix methods. In this dissertation, we develop capacitance matrix methods for exterior boundary value problems for the Helmholtz equation, a second-order elliptic PDE which governs time-harmonic wave propagation. Solutions of exterior Helmholtz problems must satisfy an asymptotic boundary condition at infinity in order to be uniquely determined. We incorporate this boundary condition into the discretization by posing the Dirichlet-to-Neumann (DtN) condition, an exact non-local boundary condition, on a circular artificial boundary. A fast Helmholtz solver of complexity O(nm log n) can then be obtained for the resulting discrete problem on an m \Theta n grid when the underlying computational domain is an an..