PRECONDITIONING OF THE P-VERSION OF THE FINITE-ELEMENT METHOD

The p-version finite element method for linear, second-order elliptic equations in an arbitrary, sufficiently smooth (incl. polygonal), boun ded domain is studied in the framework of the Domain Decomposition (DD ) method. Two types of square reference elements are used with coordin ate functions given by the products of the integrated Legendre polynom ials. Estimates for the condition numbers and some useful inequalities are given. We consider preconditioning of the problems arising on sub domains and of the Schur complement, as well as the derivation and ana lysis of the DD preconditioner for the entire system. This is done for a class of curvilinear finite elements. We obtain several DD precondi tioners 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 proo fs omitted can in most cases be found in [19,20,27].