Approximation theory for the p-version of the finite element method in three dimensions. Part 1: Approximabilities of singular functions in the framework of the Jacobi-weighted Besov and Sobolev spaces

摘要

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 a 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 vertex singularity, the edge singularity, and the vertex-edge singularity, and to explore their best approximabilities in terms of these spaces. In the forthcoming part 2, 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.