A narrow band finite element method for the level set equation

TL;DR

This paper introduces a narrow band finite element method for solving the level set equation, combining an efficient extension procedure with discontinuous Galerkin discretization and BDF time-stepping, supported by stability and accuracy analysis.

Contribution

It presents a novel extension procedure based on finite element projections and ghost-penalty, enhancing computational efficiency and enabling rigorous error analysis for level set interface tracking.

Findings

Method is computationally efficient due to narrow band approach.

Extension procedure is stable and accurate as shown by analysis.

Numerical experiments demonstrate the method's effectiveness.

Abstract

A finite element method is introduced to track interface evolution governed by the level set equation. The method solves for the level set indicator function in a narrow band around the interface. An extension procedure, which is essential for a narrow band level set method, is introduced based on a finite element $L^2$- or $H^1$-projection combined with the ghost-penalty method. This procedure is formulated as a linear variational problem in a narrow band around the surface, making it computationally efficient and suitable for rigorous error analysis. The extension method is combined with a discontinuous Galerkin space discretization and a BDF time-stepping scheme. The paper analyzes the stability and accuracy of the extension procedure and evaluates the performance of the resulting narrow band finite element method for the level set equation through numerical experiments.