Superconvergence of triquadratic finite elements for the second-order elliptic equation with variable coefficients

Abstract

This study focuses on superconvergence of the tensor-product quadratic finite element (so-called triquadratic finite element) in a regular family of rectangular partitions of the domain for the second-order elliptic equation with variable coefficients in three dimensions. In this paper, we first introduce a variable coefficients elliptic boundary value problem and its finite elements discretization, as well as some important functions such as the discrete Green's function and discrete derivative Green's function. Then, an interpolation operator of project type is given, by which we derive a interpolation fundamental estimate (so-called weak estimate) for general variable coefficients elliptic equations. Finally, combining the weak estimate and estimates for the discrete Green's function and discrete derivative Green's function, we get superconvergence estimates for derivatives and function values of the finite element approximation in the pointwise sense of the $L^{\infty }$-norm.