Iterative substructuring algorithms for the p-version finite element method for elliptic problems

In this thesis, we study iterative substructuring methods for linear elliptic problems approximated by the $p$-version finite element method. They form a class of nonoverlapping domain decomposition methods, for which the information exchange between neighboring subdomains is limited to the variables directly associated with the interface, i.e. those common to more than one subregion. Our objective is to design algorithms in $3D$ for which we can find an upper bound for the {\it condition number} $\kappa$ of the preconditioned linear system, which is independent of the number of subdomains and grows slowly with $p$. Iterative substructuring methods for the $h-$version finite element, and spectral elements have previously been developed and analysed by several authors. However, some very real difficulties remained when the extension of these methods and their analysis to the $p-$version finite element method were attempted, such as a lack extension theorems for polynomials. The corresponding results are well known for Sobolev spaces, but their extension to finite element spaces is quite intricate. In our technical work, we use and further develop extension theorems for polynomials in order to prove bounds on the condition numbers of several algorithms. We have also made many numerical tests. We can use our programs for several purposes. Not only can we compute the condition numbers and study the rate of convergence for a variety of the algorithms that we have developed, but we can also compute the bounds on these condition numbers, as given by the theory. This is useful because the theory predicts the order of magnitude and the asymptotic growth of these bounds, not of the actual condition numbers.