An immersed $CR$-$P_0$ element for Stokes interface problems and the optimal convergence analysis

TL;DR

This paper introduces an immersed finite element method based on the $CR$-$P_0$ element for Stokes interface problems with discontinuous viscosity, providing optimal convergence analysis and error estimates independent of interface position.

Contribution

It develops a novel immersed finite element method using $CR$-$P_0$ elements for interface problems, with rigorous proof of stability, optimal approximation, and error estimates.

Findings

The IFE basis functions are unisolvent on arbitrary interface elements.

The IFE spaces achieve optimal approximation capabilities.

Error estimates are independent of interface location.

Abstract

This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional $CR$-$P_0$ element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary interface elements and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.