Mathematical Sciences: Boundary Conditions for Linear Wave Propagation on Unbounded Domains

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9404488 Meade The proposer will develop accurate and efficient numerical solution procedures for problems arising from the scattering of electromagnetic waves by an inhomogeneous body. The natural domain for these problems is unbounded --- the exterior of the scatterer. This work will identify approximate reformulations of the problem on a bounded domain which can be solved by Galerkin methods, in particular the finite element method. The general idea is to truncate the unbounded domain through the selection of an appropriate artificial boundary and a corresponding boundary condition. An exact boundary condition can be used, but its non-local temporal and/or spatial dependence is an obstacle to efficient numerical computations. The primary objective of this project is to find approximate boundary conditions which are defined locally, can be implemented numerically, and yield accurate approximations to the exact solution of the scattering problem. A major challenge will be to develop the mathematical tools which can be used to balance the competing constraints of computational complexity and the different approximation errors. When an electromagnetic signal hits an obstacle, one portion of the wave continues into the object while another part of the wave is reflected back into the surrounding medium. The goal of this project is to devise accurate and efficient numerical algorithms for determining the different components of the electromagnetic wave. The mathematical statement of this problem involves the solution of partial differential equations on an unbounded domain. The fact that the domain is infinite complicates the search for an efficient numerical solution of this problem. The purpose of this research is to identify approximate reformulations of the problem which can be solved numerically without significantly degrading the quality of the computed solution. Initial studies will be directed towards the two-dimensional scalar problem. The ulti mate objective is to be able to solve the vector-valued problem in three-dimensional space. The numerical solution will call for the solution of a system of hundreds of thousands, if not millions, of equations. For this reason, the proposer will be developing codes which can be used on high-performance parallel computers. The results from this project can also be applied in a variety of other applications, including interface problems in elasticity and the scattering of acoustic, seismic, and water waves.