On the preconditioning in the domain decomposition technique for the p-version finite element method. Part I

On the preconditioning in the domain decomposition technique for the p-version nite element method. Part I P-version nite element method for the second order elliptic equation in an arbitrary suciently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (log p)² The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element