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

Xuewei Cui; Xuehai Huang

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arXiv · 2506.20240

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

Xuewei Cui·Xuehai Huang

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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.

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BibTeX RIS

Xuewei Cui, Xuehai Huang. 2025-12-04. Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem. https://arxiv.org/abs/2506.20240

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