Non-conforming finite elements on polytopal meshes

TL;DR

This paper introduces a flexible framework for non-conforming finite elements on polygonal and polyhedral meshes, including a new method called LEPNC, with theoretical analysis and numerical validation for elliptic problems.

Contribution

It develops a general functional framework for non-conforming polytopal finite elements, introduces the LEPNC method, and demonstrates their integration into the gradient discretization framework for error analysis.

Findings

The LEPNC method achieves accurate approximations on polytopal meshes.

Mass-lumping techniques improve efficiency for degenerate elliptic equations.

Numerical tests confirm the effectiveness of the proposed methods.

Abstract

In this work we present a generic framework for non-conforming finite elements on polytopal meshes, characterised by elements that can be generic polygons/polyhedra. We first present the functional framework on the example of a linear elliptic problem representing a single-phase flow in porous medium. This framework gathers a wide variety of possible non-conforming methods, and an error estimate is provided for this simple model. We then turn to the application of the functional framework to the case of a steady degenerate elliptic equation, for which a mass-lumping technique is required; here, this technique simply consists in using a different --piecewise constant-- function reconstruction from the chosen degrees of freedom. A convergence result is stated for this degenerate model. Then, we introduce a novel specific non-conforming method, dubbed Locally Enriched Polytopal Non-Conforming (LEPNC). These basis functions comprise functions dedicated to each face of the mesh (and associated with average values on these faces), together with functions spanning the local $\mathbb{P}^1$ space in each polytopal element. The analysis of the interpolation properties of these basis functions is provided, and mass-lumping techniques are presented. Numerical tests are presented to assess the efficiency and the accuracy of this method on various examples. Finally, we show that generic polytopal non-conforming methods, including the LEPNC, can be plugged into the gradient discretization method framework, which makes them amenable to all the error estimates and convergence results that were established in this framework for a variety of models.