A $\mathcal{CR}$-rotated $Q_1$ nonconforming finite element method for Stokes interface problems on local anisotropic fitted mixed meshes

TL;DR

This paper introduces a new nonconforming finite element method using a rotated $Q_1$ element on anisotropic meshes for solving Stokes interface problems, ensuring stability and optimal convergence without stabilization.

Contribution

It develops a novel rotated $Q_1$ element that remains stable on degenerate quadrilaterals and proves inf-sup stability for the combined velocity-pressure space on anisotropic meshes.

Findings

Stable on degenerate quadrilaterals

Optimal convergence order achieved

Numerical examples verify theoretical results

Abstract

We propose a new nonconforming finite element method for solving Stokes interface problems. The method is constructed on local anisotropic mixed meshes, which are generated by fitting the interface through simple connection of intersection points on an interface-unfitted background mesh, as introduced in \cite{Hu2021optimal}. For triangular elements, we employ the standard $\mathcal{CR}$ element; for quadrilateral elements, a new rotated $Q_1$-type element is used. We prove that this rotated $Q_1$ element remains unisolvent and stable even on degenerate quadrilateral elements. Based on these properties, we further show that the space pair of $\mathcal{CR}$-rotated $Q_1$ elements (for velocity) and piecewise $P_0$ spaces (for pressure) satisfies the inf-sup condition without requiring any stabilization terms. As established in our previous work \cite{Wang2025nonconforming}, the consistency error achieves the optimal convergence order without the need for penalty terms to control it. Finally, several numerical examples are provided to verify our theoretical results.