On Domain Decomposition Preconditioning in the Hierarchical P-version of the Nite Element Method

The p-version nite element method for linear, second order elliptic equations in an arbitrary, suuciently 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 polynomials for coordinate functions. Estimates for the condition numbers are given, preconditioning of the problems arising on subdomains and of the Schur complement, the derivation of the DD pre-conditioner are all considered. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p) 3) to O(1). The paper consists of six sections. We give some preliminary results for the 1D case, condition number estimates and some inequalities for the 2D reference element. The preconditioning of the Schur complement is detailed for the 2D reference element, curvilinear nite elements are considered next, and the DD preconditioning of the entire stiiness matrix is introduced and analyzed. Some related reference elements using Chebyshev nodal bases on the boundary are introduced and analyzed with respect to the eeect on the preconditioner { the (log p) 3 factor may thus be removed.