Implementation of Non-Overlapping Domain Deomposition Techniques for FETI Methods.

Abstract

In this work, some improvements to algorithms for FETI (Finite Element Tearing and Interconnecting) methods together with theoretical background are presented. A smaller part of the text is devoted to provide an optimal domain decomposition for FETI method, which is essential for the further analysis. The main part of this work is given to understand the meshes arising during discretization of numerical problems from graph theory point of view and to analyze these meshes as a graphs. This enables the use of some techniques of spectral graph theory directly applied to meshes of numerical problems. The particular purpose of this analysis is to find certain nodes in meshes to reduce numerical instability in Cholesky-SVD method, especially when applied to solving discretized version of Neumann problem (stabilizing action of general inverse of semidefinite stiffness matrix). Partly guided by intuition about solid mechanics (vibration modes), it is natural to expect the so-called “fixing nodes” near the “center” of a graph. Several candidates to these nodes are provided based on (spectral) graph techniques together with experimentally results. One of the assets of this work is the theoretical numerical analysis of one of these candidates confirming the idea that this candidate provides the best solution to the given problem, according to its definition.