FETI Algorithms for Mortar Methods

研究者Dan Stefanica将有限元撕裂和互连（FETI）算法扩展到砂浆方法，从而利用砂浆元件固有的灵活性。他建立了所得到的算法保持了fei方法对一致性元素的收敛性。这与低阶迫击炮有限元FETI算法的大量数值实验结果一致。他检查了为迫击炮的FETI算法获得的条件数估计是否清晰。他将FETI方法扩展到光谱方法和砂浆谱元，并对这些算法进行了数值和理论的收敛分析。最后，他设计了几何一致性和几何非一致性砂浆方法的双原始FETI算法。本项目中分析的算法是更大的域分解方法家族的一部分，域分解方法是一种强大、快速、易于并行化的方法，用于解决大量实际应用中出现的偏微分方程的数值解。研究者将其中一种方法，即FETI方法，与一种通用的方程离散化方法，即砂浆方法相结合。因此，他能够解决非常复杂的几何问题，同时将大部分计算精力集中在解决问题的关键部分上。FETI方法已经在大量并行代码中实现，适用于广泛的应用。该项目所开发的改进将对航空航天设计、计算力学和流体流动问题等应用具有重要意义。

The investigator, Dan Stefanica, extends the Finite Element Tearing and Interconnecting (FETI) algorithms to mortar methods, thus taking advantage of the inherent flexibility of the mortar elements. He establishes that the resulting algorithms preserve the convergence properties of the FETI methods for conforming elements. This agrees with extensive numerical experiments for FETI algorithms for low order mortar finite elements. He checks whether the condition number estimates obtained for the FETI algorithms for mortars are sharp. He extends the FETI method to spectral methods and mortar spectral elements, by providing numerical and theoretical convergence analysis for these algorithms. Finally, he designs Dual--Primal FETI algorithms for both geometrically conforming and geometrically nonconforming mortar methods. The algorithms analyzed in this project are part of the larger family of domain decomposition methods, which are powerful, fast, and easily parallelizable methods for the numerical solution of partial differential equations arising from a large spectrum of practical applications. The investigator couples one of these methods, the FETI method, with a versatile discretization of the equations, called the mortar method. Thus, he is able to solve problems with very complicated geometry while concentrating most of the computational effort on resolving the critical parts of the problem. The FETI method has already been implemented in huge parallel codes for a large spectrum of applications. Improvements in it developed by this project will be important for such applications as aerospace design, computational mechanics and fluid flow problems.