Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

TL;DR

This paper investigates how distorted triangulations affect the error bounds of nonconforming P1 finite element methods for second-order elliptic problems, revealing essential dependence on triangle angles and potential convergence issues.

Contribution

It demonstrates that, unlike conforming elements, nonconforming P1 elements' error estimates depend critically on triangulation distortion, with explicit examples showing possible slow or non-convergence.

Findings

Error bounds depend on maximum triangle angles in distorted meshes.

Examples show nonconforming P1 projections can fail to converge.

Deterioration of consistency error explains the dependence.

Abstract

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete $H^1$ norm best approximation error estimates for $H^2$ functions hold for arbitrary triangulations. However, similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on the example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily slow speed. The results complement analogous findings for conforming P1 elements.