$P_1$--nonconforming polyhedral finite elements in high dimensions

TL;DR

This paper introduces a simple, cost-effective $P_1$--nonconforming polyhedral finite element method for high-dimensional elliptic problems, using domain triangulation into parallelotes instead of simplices, achieving optimal convergence.

Contribution

It presents a novel nonconforming finite element approach for high dimensions based on polyhedral meshes, which is simpler and more efficient than traditional methods.

Findings

Achieves optimal order convergence for second-order elliptic problems.

Uses domain triangulation into parallelotes, simplifying mesh generation.

Applicable to high-dimensional elliptic problems with efficient computation.

Abstract

We consider the lowest--degree nonconforming finite element methods for the approximation of elliptic problems in high dimensions. The $P_1$--nonconforming polyhedral finite element is introduced for any high dimension. Our finite element is simple and cheap as it is based on the triangulation of domains into parallelotes, which are combinatorially equivalent to $d$--dimensional cube, rather than the triangulation of domains into simplices. Our nonconforming element is nonparametric, and on each polytope it contains only linear polynomials, but it is sufficient to give optimal order convergence for second--order elliptic problems.