A narrow band finite element method for the level set equation

TL;DR

This work develops a narrow-band finite element framework for tracking interfaces via the level set equation $\dfrac{\partial \phi}{\partial t} + \boldsymbol{u}\cdot\nabla\phi = 0$, solving $\phi$ in a thin tubular neighborhood around the evolving interface and extending it to the next transport step using a projection-based ghost-penalty extension. The method combines discontinuous Galerkin discretization in space with BDF time-stepping, and introduces two extension variants ($L^2$ and $H^1$ projections) stabilized by volumetric ghost penalties, with rigorous error bounds and long-time stability analysis. The authors provide a detailed algorithm for the narrow-band scheme, including construction of inflow boundary data, domain choices for the projection, and a validation suite across 2D/3D deforming geometries, rotating bodies, and strongly deformed spheres. Key findings show optimal convergence rates for the extension (and near-optimal surface tracking) without reinitialization, and demonstrate robustness to large deformations, while pointing to future work in adaptivity and full method-wide error analysis. Overall, the approach offers a flexible, higher-order capable framework for EF-level set computations in moving-interface problems, with solid theoretical underpinnings for the extension step and practical validation through numerical experiments.

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.

Figures (19)

• Figure 2.1: Sketch of narrow band $\mathcal{O}(\Gamma(t))=\Omega_\epsilon(t)$ and successive time slabs $\Omega_\Gamma^n \times [t_n,t_{n+1}]$, $\Omega_\Gamma^{n+1} \times [t_{n+1},t_{n+2}]$
• Figure 4.1: Adjacent Triangles $T$ and $\tilde{T}$ sharing the face $F$.
• Figure 5.1: $\Omega_h^p$, $\Omega_h^e$, and the corresponding faces $\mathcal{F}\xspace_h^{\text{GP}}$.
• Figure 5.2: Ghost penalty extension of the level set function on one extension layer, $\Omega_h^e = \mathcal{N}\xspace^1(\Omega_h^p)$.
• Figure 5.3: Ghost penalty extension of the level set function on two extension layers, $\Omega_h^e = \mathcal{N}\xspace^2(\Omega_h^p)$.

Theorems & Definitions (11)

• Remark 4.1
• Lemma 5.1
• proof
• Theorem 5.2
• proof
• Theorem 5.3
• proof
• Remark 5.1
• Lemma 5.4
• proof