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

TL;DR

This paper introduces a new nonconforming finite element method for elliptic interface problems on locally anisotropic meshes, providing improved error analysis and confirmed by numerical experiments.

Contribution

It develops a novel consistency error analysis for nonconforming elements that removes the quasi-regularity assumption, enhancing theoretical understanding.

Findings

Numerical results confirm the convergence rates.

The method demonstrates robustness and accuracy.

Error estimates are established on anisotropic meshes.

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.