Laplace domain methods for the construction of transparent boundary conditions for time-harmonic problems

We consider Helmholtz-type equations with inhomogeneous exterior domains, e.g.~exterior domains containing wave guides, which arise in the simulation of photonic devices. Usually, Sommerfeld's radiation condition is not valid for such problems. We present an alternative radiation condition called pole condition, which has been formulated by Frank Schmidt. Roughly speaking, it says that the Laplace transform of a radiating solution along a family of rays tending to infinity has a holomorphic extension to the lower half of the complex plane. To justify the validity of this condition, we show that it is equivalent to Sommerfeld's radiation condition for bounded obstacle scattering problems. Moreover, we show that for scattering problems by rough surfaces and for wave guide problems, the pole condition is also equivalent to the standard radiation conditions used in these fields.

For the numerical solution of scattering problems by finite element methods, a mesh termination problem has to be solved: What kind of boundary condition can be imposed on the artificial boundary of the computational domain such that the computed solution is not contaminated by spurious reflections at the artificial boundary? Such boundary conditions are called transparent. We describe a construction of exact transparent boundary conditions based on the pole condition which does not rely on the explicit knowledge of a fundamental solution or a series representation of the solution. There exists an explicit formula expressing the exterior solution in a stable way in terms of quantities defined in the Laplace domain. This provides an efficient numerical method for the evaluation of the exterior solution and distinguishes the proposed method from other methods, e.g. the Perfectly Matched Layer (PML) method. The total computational cost is typically dominated by the finite element solution of the interior problem.