Postdoc: Iterative Methods Arising in PDE's

本计画有三个研究主题，分别是二阶偏微分方程离散化所产生的非对称线性方程组的迭代求解方法。 其中，两个集中在时间谐波传播和对流扩散问题之一。 第一个主题包括扩展算法的解决方案的时间谐波散射问题的亥姆霍兹方程，从二维到三维。 基于使用精确辐射条件的离散化-因此允许相对于散射体的小计算域-这些算法使用与快速亥姆霍兹求解器相结合的嵌入技术，从而为变得越来越重要的应用程序提供快速且可并行化的算法。 第二个主题是最近的代数多层技术应用于外部亥姆霍兹问题的调查。 这些技术可以导致解决这类问题的最佳复杂性的算法。 最后，第三个主题涉及研究对流扩散问题离散化的稳定化技术对离散问题迭代解的影响。

This project has three research topics from the field of iterative solution methods for nonsymmetric linear systems of equations arising in the discretization of second-order PDEs. Of these, two focus on time-harmonic wave propagation and one on convection-diffusion problems. The first topic consists in extending algorithms for the solution of time-harmonic scattering problems for the Helmholtz equation, from 2D to 3D. Based on a discretization which uses an exact radiation condition - and hence permits small computational domains relative to the scatterer - these algorithms use an imbedding technique combined with a fast Helmhotz solver, resulting in fast and well-parallelizable algorithms for an application which is becoming increasingly important. The second topic is an investigation of recent algebraic multilevel techniques applied to exterior Helmholtz problems. These techniques could lead to an algorithm of optimal complexity for solving this type of problem. Finally, the third topic involves investigating effects of stabilization techniques for discretization of convection-diffusion problems on iterative solution of discrete problems.