APPROXIMATION THEORY FOR THE P-VERSION OF THE FINITE ELEMENT METHOD IN THREE DIMENSIONS IN THE FRAMEWROK OF JACOBI-WEIGHTED BESOV SPACES Part I : Approximabilities of singular functions Dedicated to Professor Ivo Babuška on the occasion of his 80-th birthday

This paper is the first in a series devoted to the approximation theory of the p-version of the finite element method in three dimensions. In this paper, we introduce the Jacobi-weighted Besov and Sobolev spaces in the three-dimensional setting and analyze the approximability of functions in the framework of these spaces. In particular, the Jacobi-weighted Besov and Sobolev spaces with three different weights are defined to precisely characterize the natures of the vertexsingularity, the edge singularity and vertex-edge singularity, and to explore their best approximabilities in terms of these spaces. In the forth coming Part II, we will apply the approximabilities of these singular functions to prove the optimal convergence of the p-version of the finite element method for elliptic problems in polyhedral domains, where the singularities of three different types occur and substantially govern the convergence of the finite element solutions.