Nodal $\mathcal{O}(h^4)$-Superconvergence in 3D by Averaging Piecewise Linear, Bilinear, and Trilinear FE Approximations

Abstract

We construct and analyse a nodal $\mathcal{O}(h^4)$-superconvergent FE scheme for approximating the Poisson equation with homogeneous boundary conditions in three-dimensional domains by means of piecewise trilinear functions. The scheme is based on averaging the equations that arise from FE approximations on uniform cubic, tetrahedral, and prismatic partitions. This approach presents a three-dimensional generalization of a two-dimensional averaging of linear and bilinear elements which also exhibits nodal $\mathcal{O}(h^4)$-superconvergence (ultraconvergence). The obtained superconvergence result is illustrated by two numerical examples.