Mathematical Sciences: Effective Numerical Solution of Elliptic Equations

该项目将支持研究有效和稳健的 解决大型线性系统的策略， 从椭圆边值问题的离散化。 主要重点将放在非自伴的解决方案， 不确定的问题，例如，外部亥姆霍兹问题 两 预处理策略和计算策略 将研究并行机。 领域的 预处理策略，目标将是推导 条件数的估计以及 特征值和奇异值的预处理系统。 伪谱离散的有限差分预处理 差分或有限元离散化也将被 研究了 在计算策略领域， 各种预处理有效性和实施 将研究SIMD和MIMD计算机系统的策略。 在求解椭圆型方程边值问题时 在数值上，有必要解决非常大的系统， 当连续问题是 被一个离散的问题所取代。 主要研究者 将开发既有效又可靠的数值方法 解决这样的系统。

This project will support the study of effective and robust strategies for the solution of large linear systems that arise from the discretization of elliptic boundary value problems. Major emphasis will be on the solution of non-self-adjoint, indefinite problems, e.g., the exterior Helmholtz problem. Both preconditioning strategies and computational strategies on parallel machines will be investigated. In the area of preconditioning strategies, the objective will be the derivation of estimates of condition numbers as well as the distribution of eigenvalues and singular values of the preconditioned systems. The preconditioning of pseudospectral discretizations by finite difference or finite element discretizations will also be studied. In the area of computational strategies, the effectiveness and implementation of various preconditioning strategies on SIMD and MIMD computer systems will be studied. In solving boundary value problems for elliptic equations numerically, it becomes necessary to solve the very large systems of linear equations that arise when the continuous problem is replaced with a discrete problem. The principal investigator will develop numerical methods that are both efficient and robust for solving such systems.