A modified $P_1$ - immersed finite element method

TL;DR

This paper introduces a modified P1 immersed finite element method that incorporates flux line integrals, improving accuracy and efficiency for interface problems with optimal error estimates and fewer degrees of freedom.

Contribution

The paper presents a novel P1-based IFEM with flux integral terms, reducing degrees of freedom while achieving optimal convergence, inspired by DG methods.

Findings

Achieves optimal H^1 and L^2 error estimates.

Demonstrates robustness through numerical experiments.

Uses fewer degrees of freedom than traditional DG methods.

Abstract

In recent years, the immersed finite element methods (IFEM) introduced in \cite{Li2003}, \cite{Li2004} to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence. In this paper, we propose an improved version of the $P_1$ based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional $P_1$ finite element method. We prove $H^1$ and $L^2$ error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme.