Frenet Immersed Finite Element Spaces on Triangular Meshes

TL;DR

This paper introduces Frenet-immersed finite element spaces on triangular meshes for elliptic interface problems, enabling exact interface jump conditions and demonstrating optimal approximation and convergence rates.

Contribution

It extends the Frenet-Serret based IFE framework from rectangular to triangular meshes and develops high-degree spaces with improved conditioning.

Findings

Optimal approximation capabilities demonstrated through numerical examples.

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

Extended IFE framework to triangular meshes with high-degree spaces.

Abstract

In this paper, we develop geometry-conforming immersed finite element (GC-IFE) spaces on triangular meshes for elliptic interface problems. These IFE spaces are constructed via a Frenet-Serret mapping that transforms the interface curve into a straight line, allowing the interface jump conditions to be imposed exactly. Extending the framework of [7,8] from rectangular to triangular meshes, we introduce three procedures for constructing high-degree Frenet-IFE spaces: an initial construction based on monomial bases, a generalized construction using orthogonal polynomials, and reconstruction methods aimed at improving the conditioning of the associated mass matrix. The optimal approximation capability of the proposed IFE spaces is demonstrated through numerical examples. We further incorporate these spaces into interior penalty discontinuous Galerkin methods for elliptic interface problems and observe optimal convergence rates in the $H^1$ and $L^2$ norms.