A Nonconforming Finite Element Method for Elliptic Interface Problems on Locally Anisotropic Meshes

TL;DR

This work develops a nonconforming $P_1$ finite element method for elliptic interface problems on locally anisotropic, interface-fitted hybrid meshes, enabling accurate resolution of discontinuities across material interfaces. A key contribution is a novel consistency error analysis that removes the usual quasi-regularity assumption and leverages a directional decomposition alongside vertex-based interpolation to achieve optimal-order estimates on anisotropic elements. The authors prove an $O(h)$ convergence rate in the energy-like norm via a Strang-type argument, and substantiate the theory with numerical tests on circular and flower-shaped interfaces, including scenarios with varying interface positions. The method integrates gracefully with standard nonconforming elements: it reduces to the Crouzeix–Raviart element on triangles and to the Park–Sheen element on quadrilaterals, offering a robust and flexible tool for complex interface geometries in practical applications.

Abstract

We propose a new nonconforming \(P_1\) finite element method for elliptic interface problems. The method is constructed on a locally anisotropic mixed mesh, which is generated by fitting the interface through a simple connection of intersection points on an interface-unfitted background mesh, as introduced in \cite{Hu2021optimal}. We first establish interpolation error estimates on quadrilateral elements satisfying the regular decomposition property (RDP). Building on this, the main contribution of this work is a novel consistency error analysis for nonconforming elements, which removes the quasi-regularity assumption commonly required in existing approaches. Numerical results confirm the theoretical convergence rates and demonstrate the robustness and accuracy of the proposed method.

Figures (12)

• Figure 1: Geometric interface and mesh interaction: (a) the computational domain for the interface problem; (b) unfitted mesh $\mathcal{T}_h$; (c) local anisotropic hybrid mesh $\tilde{\mathcal{T}}_h$.
• Figure 2: The interface macro element
• Figure 3: Degrees of freedom
• Figure 4: The reference quadrilateral element $\hat{Q}$
• Figure 5: triangle element satisfying Maxac.

Theorems & Definitions (21)

• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• Lemma 3.4: Trace theorem
• Lemma 3.5: Acosta and Durán, 2000
• proof
• Lemma 3.6
• proof
• Theorem 3.7