Mathematical Sciences: Algorithms and Numerical Analysis forPartial Differential Equations

该奖项将支持主要研究人员继续研究偏微分方程数值解的有限元方法。特别是，研究人员将构建预条件条件，使他们能够更有效地求解由此产生的代数方程系统。从他们的研究中获得的理论见解应该会导致开发用于并行计算机的改进的软件，这反过来将导致改进二维和三维空间中的一系列问题的解决方案。计算机已经永远地改变了我们寻找偏微分方程解的方式，这些偏微分方程解出现在科学和技术中。不幸的是，即使是最强大的超级计算机解决复杂系统的能力也往往受到微分方程式离散化所产生的代数方程的巨大规模的限制。为了克服这种“大小”的限制，数值分析家们想出了一些方法来“预置”这些代数方程，以使它们更容易被计算机所理解。这笔赠款的主要研究人员将寻求更好的方法来实现这一预条件。

This award will support the efforts of the principal investigators to continue their studies of finite element methods for the numerical solution of partial differential equations. In particular, the researchers will construct preconditioners that will enable them to solve the resulting systems of algebraic equations more efficiently. The theoretical insights gained from their studies should lead to the development of improved software for use on parallel computers, which in turn will lead to improved solutions of a host of problems in two and three space dimensions. The computer has changed forever the way we go about finding solutions of partial differential equations that arise in science and technology. Unfortunately the ability of even the most powerful supercomputer to solve complicated systems is limited very often by the sheer size of the algebraic equations that result from discretizing the differential equations. In order to overcome this "size" limitation numerical analysts have devised ways to "precondition" these algebraic equations so as to make them more digestible to the computer. The principal investigators on this grant will seek better ways to achieve this preconditioning.