An immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes

TL;DR

This paper introduces a novel immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes, achieving optimal convergence rates through a specially constructed IFE space.

Contribution

It develops a lowest-order immersed Raviart-Thomas finite element method with modified basis functions for better accuracy on unfitted meshes, including rigorous error analysis.

Findings

Optimal convergence rates achieved on unfitted meshes

The IFE basis functions are unisolvent and have strong approximation properties

Numerical examples confirm theoretical error estimates

Abstract

This paper presents a lowest-order immersed Raviart-Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element finite (IFE) is constructed by modifying the traditional Raviart-Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal error estimates are rigorously proved for the mixed IFE method and some numerical examples are also provided to validate the theoretical analysis.