Least-Squares Finite Element Methods and Optimization-Based Domain Decomposition Methods for Partial Differential Equations

9806358贡兹堡在过去的几年里，工程界和数学界对解决流体、电磁学、弹性和其他应用中的各种问题的最小二乘有限元方法表现出越来越大的兴趣。最小二乘法的巨大前景源于这样一个事实，即与其他离散化方案相比，它们导致了更容易在计算机上解决的离散问题。在过去，PI研究了最小二乘有限元方法的许多方面。这些方法包括：在最小二乘泛函中使用依赖于网格的权重以达到最佳精度，解决为使这些方法实用和具有竞争力而需要解决的实际实施问题，以及这些方法在具有不连续系数的问题上的应用，例如，非均匀介质特性引起的问题。PI计划将最小二乘有限元方法应用于优化和控制问题，并在最小二乘设置中开发、分析和实施区域分解算法。区域分解方法由于其在并行处理环境中的实用性而引起了更多的关注。PI基于最优化或最优控制思想发展了新的非重叠区域分解方法，具有许多理想的特征，其中最重要的可能是它们很容易扩展到非线性问题。PI计划引入预处理子来加速方法的性能，查看不同的泛函和优化参数以将其分解为子域，从而再次获得更有效的算法，将算法应用于偏微分方程组的优化问题的求解和分析，开发时间相关问题的算法，并在由奔腾处理器集群组成的并行计算机上实现算法。

9806358 Gunzburger In the last few years, the engineering and mathematical communities have shown increasing interest in least-squares finite element methods for solving a variety of problems in fluids, electromagnetics, elasticity, and other applications. The great promise of least-squares methods arises from the fact that, when compared to other discretization schemes, they lead to discrete problems that are much easier to solve on a computer. In the past the PI has studied numerous facets of least-squares finite element methods. These include: the use of mesh-dependent weights in least-squares functionals in order to achieve optimally accuracy, the solution of practical implementation issues that needed to be addressed in order to make these methods practical and competitive, and the application of these methods to problems with discontinuous coefficients that arise, e.g., from inhomogeneous media properties. The PI plans to apply least-squares finite element methodologies to optimization and control problems and to develop, analyze, and implement domain decomposition algorithms in the least-squares setting. Domain decomposition methods have attracted even more attention due to their usefulness in a parallel processing environment. The PI has developed novel non-overlapping domain decomposition methods based on optimization or optimal control ideas that posses numerous desirable features, the most important perhaps being that they are easily extended to nonlinear problems. The PI plans to introduce preconditioners to speed-up the performance of the methods, to look at different functionals and optimization parameters on which to base the decomposition into subdomains so that again more efficient algorithms are obtained, to apply and analyze algorithms to the solution of optimization problems for partial differential equations, to develop algorithms for time-dependent problems, and to implement algorithms on parallel computers consisting of clusters of Pentium processors.