Mathematical Sciences: Effective Numerical Solution of Elliptic Equations

该项目将支持研究由椭圆边值问题离散化引起的大型线性系统的有效和稳健的解决策略。主要的重点将放在非自伴随的、不定的问题的解上，例如，外部亥姆霍兹问题。本文将研究并行机器上的预处理策略和计算策略。在预处理策略领域，目标将是条件数的估计的推导，以及预条件系统的特征值和奇异值的分布。本文还将研究有限差分或有限元离散对伪谱离散的预处理。在计算策略方面，将研究各种预处理策略在SIMD和MIMD计算机系统上的有效性和实施。在用数值方法求解椭圆方程的边值问题时，当连续问题被离散问题取代时，就有必要求解非常大的线性方程组。首席研究员将开发既有效又稳健的数值方法来解决这类系统。

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.