Abstract:

Finite element approximation of vector equations gives rise to very large, sparse linear systems. In this dissertation, we study some domain decomposition methods for finite element approximations of vector--valued problems, involving the curl and the divergence operators. Edge and Raviart--Thomas finite element are employed. Problems involving the curl operator arise, for instance, when approximating Maxwell's equations and the stream function--vorticity formulation of Stokes' problem, while mixed approximations of second order elliptic equations and stabilized mixed formulations of the Stoke' problem give rise to problems involving the divergence operator.

We first consider Maxwell's equations in three dimensional conductive media using implicit time--stepping. We prove that the condition number of a two-level overlapping algorithm is bounded independently of the number of unknowns, the number of subregions, and the time step.

For the same equation in two dimensions, we consider two new iterative substructuring methods. The first one is based on individual edges, while the second one is a Neumann-Neumann method. We show that the condition numbers of the corresponding methods increase slowly with the number of unknowns in each substructure, but are independent of the time step and even large jumps of the coefficients. We also analyze similar preconditioners for a three--dimensional vector problem involving the divergence operator, and prove that the preconditioners are quasi--optimal and scalable in this case as well.

For each method, we provide a series of numerical experiments, that confirm our theoretical analysis.

This work generalizes well--known results for scalar second order elliptic equations and has required the development of several new technical tools.