Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem
Xuewei Cui, Xuehai Huang
TL;DR
The paper introduces a low-order, nonconforming discretization of the smooth 3D de Rham complex, featuring a new tangentially continuous H1-nonconforming vector space Φ_h and a compatible W_h–Φ_h–V_h^div–Q_h quartet. It then derives a decoupled mixed finite element formulation for a fourth-order elliptic singular perturbation, transforming it into a sequence of second-order problems and leveraging an interpolation operator I_h^ND to achieve parameter-robust, optimal convergence without stabilization. A rigorous stability and regularity framework is established, showing equivalence to the primal problem and proving ε-uniform error estimates. Numerical experiments confirm robust performance in both standard and boundary-layer regimes, with the improved method (using I_h^ND) delivering optimal rates under strong boundary layers.
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 Nédélec 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.