The Effective Numerical Solution of Elliptic Equations

9501256本研究项目的主要工作包括两个方面：(1)研究椭圆边值问题切比雪夫谱配置方法的预处理策略；(2)研究具有实际意义的方程的一阶最小二乘方法。线弹性力学方程具有特殊的重要性和吸引力。这些问题的后验误差估计的重要性被需要更深的先验误差估计所平衡。获得先验误差估计的困难源于散度算子在这些问题中的显式出现。在研究Stokes方程的数值方法时也出现了类似的困难。流体流动和弹性力学中出现了许多重要的实际问题，这些问题必须在计算机上解决。然后，这些问题的解就是椭圆边值问题的解的近似。这一领域的大多数研究都强调寻找一个计算机问题，其解确实是椭圆型问题的解的非常好的近似值。尽管如此，人们仍然必须找到有效和相对廉价的方法来解决计算机问题。这最后一个非常现实的问题，是本项目研究的重点。

9501256 Parter The major efforts of this research project are twofold: (1) The study of preconditioning strategies for the Chebyshev spectral collocation method for elliptic boundary-value problems, and (2) the study of first order least squares methods for equations of practical import. The equations of linear elasticity are of special importance and interest. The significance of a- posteriori error estimates for these problems is balanced by the need for deeper a-priori error estimates. The difficulty in obtaining a-priori error estimates stems from the explicit appearance of the divergence operator in these problems. Similar difficulties appear in the study of numerical methods for the Stokes equations. There are many important practical problems which arise in fluid flow and elasticity which lead to problems which must be solved on the computer. The solutions of these problems are then approximations to solutions of elliptic boundary value problems. Most of the research in this area has emphasized finding a computer problem whose solution is indeed a very good approximation to the solution of the elliptic problem. Having done that, one must still find efficient and relatively cheap methods to solve the computer problem. This last, very practical problem, is the focus of the research in this project.