The Frenet immersed finite element method for elliptic interface problems: An error analysis

TL;DR

This paper provides an error analysis of the Frenet immersed finite element method for elliptic interface problems, demonstrating optimal convergence in $L^2$ and energy norms through new trace inequalities.

Contribution

It introduces a new Frenet IFE space with local conformity and establishes a trace inequality, enabling optimal convergence analysis for the method.

Findings

Proves optimal convergence in $L^2$ norm.

Proves optimal convergence in energy norm.

Establishes a critical trace inequality for Frenet IFE functions.

Abstract

This article presents an error analysis of the recently introduced Frenet immersed finite element (IFE) method. The Frenet IFE space employed in this method is constructed to be locally conforming to the function space of the associated weak form for the interface problem. This article further establishes a critical trace inequality for the Frenet IFE functions. These features enable us to prove that the Frenet IFE method converges optimally under mesh refinement in both $L^2$ and energy norms.