The Immersed Skeletal Finite Element Method for Elliptic Interface Problems

TL;DR

This paper introduces a novel immersed skeletal finite element method for elliptic interface problems that combines skeletal and standard FEM to achieve optimal convergence on unfitted meshes.

Contribution

The paper develops a new immersed finite element scheme integrating skeletal FEM with standard FEM, providing optimal convergence and flexibility for interface problems.

Findings

Achieves optimal convergence in $H^1$ and $L^2$ norms.

Demonstrates high accuracy and efficiency through numerical experiments.

Flexible scheme suitable for complex interface geometries.

Abstract

In this paper, we present a new immersed finite element scheme for solving elliptic interface problems on unfitted meshes by combining the skeletal finite element method (FEM) with the standard FEM. The skeletal FEM is used for the interface elements. In other words, we take piecewise functions as the unknowns inside the interface element and on its boundary. We employ the immersed finite element functions as interior functions that precisely satisfy the interface conditions. On the interface edges, we define two boundary functions to capture the discontinuity. The Lagrange element is used for the non-interface elements. The proposed scheme is simple and flexible. We prove that this scheme achieves optimal convergence orders in both the $H^1$ norm and $L^2$ norm. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method.