Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem

TL;DR

This paper introduces a low-order nonconforming finite element complex in three dimensions, applied to a fourth-order elliptic singular perturbation problem, achieving uniform optimal convergence without extra stabilization.

Contribution

It develops a novel discretization of a smooth de Rham complex that conforms to the classical complex while being nonconforming, and applies it to a decoupled mixed finite element method for singular perturbation problems.

Findings

Achieves uniform optimal convergence rates regardless of perturbation parameter.

Introduces a modified nodal interpolation operator for Nédélec elements.

Provides a decoupled mixed method avoiding additional stabilization.

Abstract

A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous $H^1$-nonconforming vector-valued finite element space, the lowest-order Raviart-Thomas space, and piecewise constant functions. While nonconforming for the smooth complex, the discretization conforms to the classical de Rham complex. It is applied to develop a decoupled mixed finite element method for a fourth-order elliptic singular perturbation problem, focusing on the discretization of a generalized singularly perturbed Stokes-type equation. In contrast to Nitsche's method, which requires additional stabilization to handle boundary layers, the nodal interpolation operator for the lowest-order N\'{e}d\'{e}lec element of the second kind is introduced into the discrete bilinear forms. This modification yields a decoupled mixed method that achieves optimal convergence rates uniformly with respect to the perturbation parameter, even in the presence of strong boundary layers, without requiring any additional stabilization.