The Numerical Solution of Elliptic Equations

9704852之后，这些研究将集中在三个具体的项目上：(1)一阶系统最小二乘方法，特别强调弹性问题，(2)一般的重叠网格问题，和(3)谱配置方法的预处理策略。目前，弹性力学问题的计算方法主要是基于混合方法，这导致了难以解决的不确定问题。FOSLS方法产生了通常更容易解决的大型系统。然而，尽管几个独立的小组作出了努力，但对于一般边界条件的弹性问题的有用的FOSLS公式仍然是一个主要问题。重叠网格法已被证明是解决复杂几何区域问题的有效工具。此外，对这些方法的分析一直是缺乏的，直到最近正型差分方程组的突破。由于这种不同方法的精确度有一定的限制，因此需要进一步分析。光谱配置法非常准确，但条件非常糟糕。因此，预适应策略是必不可少的。对预适应方法有超过25年的经验和研究。尽管如此，仍然需要更好的方法。此外，这些方法中的一些方法的数学证明是不完整的。这项研究的基本主题可以简单地说是：寻找有效的数值方法来解决力学和材料科学中的重要边值问题，并提供其有效性的数学证明。在弹性问题中，目前使用的大多数方法执行起来都很昂贵。因此，寻求一种新的公式，它将提供准确的近似，可以以合理的成本计算。在另外两种情况下，一种是处理既定的方法，这些方法要么难以实施，要么数学基础不完整。这项研究的目的是为了更全面地理解数学，既澄清现有的方法，又为开发新的、更有效的方法提供基础。

9704852 Parter These investigations will focus on three specific projects: (1) First Order Systems Least Square (FOSLS) methods with a special emphasis on problems in elasticity, (2) the general over-lapping grid problem, and (3) preconditioning strategies for Spectral Collocation Methods. At this time computational methods for problems in elasticity are primarily based on mixed methods which lead to indefinite problems which have proven difficult to solve. The FOSLS approach leads to large systems which are generally much easier to solve. However, despite the efforts of several independent groups, a useful FOSLS formulation of the elasticity problems for general boundary conditions is still a major problem. The overlapping grid method has proven itself an effective tool for problems set in regions with complicated geometry. Further, the analysis of these methods has been lacking until a recent breakthrough for difference equations of positive type. Since there are limitations on the order of accuracy of such difference methods there is a need for further analysis. Spectral Collocation methods are extremely accurate and very badly conditioned. Hence preconditioning strategies are essential. There are over twenty-five years of experience and study of preconditioning methods. And, still, there is a need for better approaches. Moreover, the mathematical justification for some of these methods is incomplete. The basic theme of this research can be stated simply "Find effective numerical methods to solve the important boundary-value problems of mechanics and material science, and provide mathematical proofs of their validity." In the problems of elasticity most of the methods now in use are expensive to implement. Hence, a new formulation is sought which will provide accurate approximations which can be computed at a reasonable cost. In the other two cases one is dealing with established methods which are either difficult to impleme nt or whose mathematical basis is incomplete. This research is aimed at a more complete mathematical understanding which will both clarify the existing methods and provide a basis for the development of new, more effective methods.