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

9404488米德：对于非均匀体的电磁波散射所引起的问题，提出准确而有效的数值解程序。这些问题的自然域是无界的——散射体的外部。这项工作将确定问题在有界域上的近似重新表述，可以用伽辽金方法，特别是有限元方法来解决。一般思想是通过选择合适的人工边界和相应的边界条件截断无界域。可以使用精确的边界条件，但其非局部时间和/或空间依赖性是有效数值计算的障碍。这个项目的主要目标是找到近似的边界条件，这些边界条件是局部定义的，可以在数值上实现，并产生精确的近似散射问题的精确解。一个主要的挑战将是开发数学工具，可以用来平衡计算复杂性和不同近似误差的竞争约束。当电磁信号碰到障碍物时，一部分波继续进入物体，而另一部分波被反射回周围的介质中。这个项目的目标是设计出精确和有效的数值算法来确定电磁波的不同组成部分。这个问题的数学表述涉及无界域上偏微分方程的解。定义域是无限的这一事实使寻找这一问题的有效数值解变得复杂。本研究的目的是确定问题的近似重新表述，这些近似表述可以在不显著降低计算解质量的情况下进行数值求解。最初的研究将针对二维标量问题。最终目标是能够在三维空间中解决向量值问题。数值解将要求解一个由几十万甚至几百万个方程组成的系统。因此，建议者将开发可用于高性能并行计算机的代码。该项目的结果也可以应用于各种其他应用，包括弹性界面问题和声波、地震和水波的散射。

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.