Addressing Integration Error for Polygonal Finite Elements Through Polynomial Projections: A Patch Test Connection

TL;DR

This paper introduces a polynomial projection method to improve the polynomial consistency of polygonal finite elements, ensuring they pass the patch test and achieve optimal convergence rates with moderate computational cost.

Contribution

It proposes a novel polynomial projection technique inspired by mimetic finite differences to restore polynomial consistency in polygonal finite elements.

Findings

Restores polynomial consistency and patch test satisfaction.

Achieves optimal convergence rates with moderate integration points.

Reduces computational cost to levels comparable with standard elements.

Abstract

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.