Domain Decomposition Methods: Algorithms and Theory

大规模的连续介质力学模拟通常需要非常大的线性或非线性代数方程组的有效解以及强大的计算机系统。这类问题越来越多地在并行或分布式计算系统上得到解决，这些系统提供了对大量存储器和大量快速处理器的访问。这个项目的目标是进一步开发域分解类型的求解器，这种算法尊重现代计算系统的内存层次结构，并允许获得接近峰值浮点性能的性能。对于如何调整这些算法以适应不同应用中出现的问题，已经有了很好的数学理论和很好的理解，但也确实需要进一步扩大我们的理解，改进和数值测试新的算法，特别是考虑到具有非常强大的处理器的并行计算系统的出现。这个项目的一个主要目标是进一步了解FETI-DP和BDDC家族的区域分解方法。这一领域的工作方向受到学术计算科学家与美国国家实验室和工程软件行业科学家之间的互动的强烈影响。与桑迪亚-阿尔伯克基实验室的联系尤其强烈。领域分解方法在这些实验室的非常大型和重要的应用中得到了广泛的应用，并且领域分解方法也在软件系统中取得了重大进展，这些软件系统被大量的工程师用于设计和检查机械部件和更大的工程结构。特别重要的应用领域是弹性和电磁学。后者有助于为蜂窝电话系统、其他无线通信系统以及微电子的改进奠定基础。

Large scale continuum mechanics simulations routinely requirethe efficient solution of very large linear or nonlinear systemsof algebraic equations as well as powerful computer systems.Such problems are increasingly solved on parallel or distributedcomputing systems which provide access to a large amount of memoryas well as a substantial number of fast processors. The goal ofthis project is to further develop solvers of domain decompositiontype, algorithms which respect the memory hierarchy of modern computing systems and allow for a performance which approaches peak floating pointperformance. There already exists a well developed mathematical theory and a good understanding of how to adjust these algorithms to problemsarising in different applications but there is also a real needto further extend our understanding and to improve and numericallytest new algorithms especially in view of emergence of parallelcomputing systems with very many powerful processors. A maingoal of this project is the further understanding of theFETI-DP and BDDC families of domain decomposition methods.The direction of the work in this field is strongly influencedby the interaction between academic computational scientistsand scientists at the US national laboratories and inthe engineering software industry. The connection to the SANDIA-Albuquerque laboratories is particularly strong. Domain decompositionmethods are used extensively in very large and important applicationsat these laboratories and the domain decomposition methods arealso making significant inroads in software systems that are usedby a large community of engineers for the design and checkingof machine parts and larger engineering structures. Particularly importantapplication areas are elasticity and electromagnetics. The latterfield helps create a foundation for improvements of cellularphone systems, other wire-less communication systems, as well asmicro electronics.