Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations

We consider three problems for the Helmholtz equation in interior and exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee comments and added several reference

Time Domain Analysis of Electromagnetic Scattering From Multiple Cavities Embedded in a Ground Plane

This work examines the scattered fields produced when a transient wave is reflected from an infinite perfect electric conductor (PEC) ground plane with multiple embedded cavities. Incident and reflected waves will be decomposed into transverse magnetic to the z direction (TMz) and transverse electric to the z direction (TEz) polarizations, with primary focus given to the TMz. Cavities may be unfilled, partially filled, or fully filled with non-magnetic dielectric material and no assumptions are made regarding similarity, regularity, or periodicity. The Newmark method is used to discretize time and a variational formulation is presented for each time step. The principle outcome is to show that the variational formulation of the scalar problem is well posed. Additionally, the variational formulation is applied in a stable numerical model using the finite element-boundary integral (FE-BI) method. Interior fields are approximated using the finite element method (FEM) for each time step, then the boundary integral is applied using the appropriate Green’s function to approximate exterior scattered fields. The exterior fields for one time step provide the boundary conditions for the interior problem at the next time step. In this way, the numerical model marches through time. Various numerical experiments are run to examine the effect of coupling on aperture and external fields. Of particular interest are the differences between single-cavity and multiple-cavity solutions