Preconditioning of the p-version of the finite element method Original Research Article

The p-version finite element method for linear, second-order elliptic equations in an arbitrary, sufficiently smooth (incl. polygonal), bounded domain is studied in the framework of the Domain Decomposition (DD) method. Two types of square reference elements are used with coordinate functions given by the products of the integrated Legendre polynomials. Estimates for the condition numbers and some useful inequalities are given. We consider preconditioning of the problems arising on subdomains and of the Schur complement, as well as the derivation and analysis of the DD preconditioner for the entire system. This is done for a class of curvilinear finite elements. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p)3) to O(1). This paper is based on [19–21,27]. We have omitted most of the proofs in order to shorten it and have described instead what could be done as well as outlined some additional ideas. The full proofs omitted can in most cases be found in [19,20,27].