616 DOMAIN DECOMPOSITION ALGORITHMS FOR THE P-VERSION FINITE ELEMENT
METHOD FOR ELLIPTIC PROBLEMS L. Pavarino, September 1992
Domain decomposition algorithms based on the Schwarz framework were originally proposed for the h-version finite element method for elliptic problems. In this thesis, we study some Schwarz algorithms for the p-version finite element method, in which increased accuracy is achieved by increasing the degree p of the elements while the mesh is fixed. These iterative algorithms, often of conjugate gradient type, are both parallel and scalable, and therefore very well suited for massively parallel computing.
We consider linear, scalar, self adjoint, second order elliptic problems and quadrilateral elements in the finite element discretization. For a class of overlapping methods, we prove a constant bound, independent of the degree p, the mesh size H and the number of elements N , for the condition number of the iteration operator. This optimal result holds in two and three dimensions for additive and multiplicative schemes, as well as variants on the interface.
We consider then local refinement for the same class of overlapping methods in two dimensions. Optimal bounds are obtained under certain hypothesis on the choice of refinement points, while in general almost optimal bounds with logarithmic growth in p are obtained. In the analysis of these local refinement methods, we prove some results of independent interest, such as a polynomial discrete Sobolev inequality and a bounded decomposition of discrete harmonic polynomials.
Iterative substructuring methods in two dimensions are also considered. We use the additive Schwarz framework to prove almost optimal bounds as in the h-version finite element method.
Results of numerical experiments, confirming the theoretical results, are conducted in two dimensions for model problems.