Domain decomposition and preconditioning in the hierarchical p-version finite element method

Prof. Vadim Korneev

Department of Mathematics of UMBC and Department of Matematics and Mechanics of St.-Petersburg State University

The p-version finite element method for linear second order elliptic equations in an arbitrary, sufficiently smooth domain is studied in the framework of the domain decomposition (DD) method. Two types of square reference elements are used with the products of the integrated Legendre polinomials taken for coordinate functions. Estimates for the condition numbers are given, preconditioners for the problems arising on subdomains and for the Shur complement and the derivation on their basis of the global preconditioner are all considered. We obtained several DD preconditioners for which the generalized condition numbers vary from O((log p)^3) to O(1). All preconditioners lead to relatively easy and cheap iterative algorithms.