Postdoc: Iterative Methods Arising in PDE's

本课题在二阶偏微分方程组的离散化过程中产生的非对称线性方程组的迭代求解方法领域有三个研究课题。其中，两个研究了时间谐波传播问题，一个研究了对流扩散问题。第一个主题是将Helmholtz方程的时间调和散射问题的求解算法从2D扩展到3D。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较小的计算区域-这些算法使用嵌入技术与快速Helmhotz求解器相结合，从而为日益重要的应用产生快速且可很好地并行化的算法。第二个主题是对最近应用于Helmholtz外问题的代数多能级技术的研究。这些技术可能导致解决这类问题的最优复杂性的算法。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.