Numerical surgery for mean curvature flow of surfaces (2210.14046v3)

Abstract: A numerical algorithm for mean curvature flow of closed mean convex surfaces with surgery is proposed. The method uses a finite element based mean curvature flow algorithm based on a coupled partial differential equation system which directly provides an approximation for mean curvature and outward unit normal. The proposed numerical surgery process closely follows the analytical surgery of Huisken & Sinestrari, and Brendle & Huisken. The numerical surgery approach is described in detail, along with extensions to other geometric flows and methods. Numerical experiments report on the performance of the numerical surgery process.

• The paper introduces a numerical surgery technique to excise high curvature regions, thereby enabling continued simulation past singularities.
• The paper employs a finite element-based algorithm with proven H1-norm error bounds up to the singularity threshold.
• Extensive numerical experiments validate the method's ability to handle complex topology changes in evolving surfaces.

Numerical Surgery for Mean Curvature Flow of Surfaces: A Detailed Analysis

The paper presents a novel numerical algorithm for the mean curvature flow (MCF) of closed mean convex surfaces, particularly incorporating a surgery mechanism reminiscent of analytical approaches by Huisken & Sinestrari and Brendle & Huisken. This work extends previous theoretical frameworks of MCF into the computational domain, providing insights into the numerical handling of topology changes in evolving surfaces.

The proposed methodology employs a finite element-based MCF algorithm, leveraging a system of coupled partial differential equations to approximate mean curvature and outward unit normal vector fields. The introduction of numerical surgery aims to tackle the singularity issues associated with high curvature regions on surfaces during flow evolution. The process halts the flow as it approaches a singularity (marked by exceeding a mean curvature threshold), excises the high curvature regions, and replaces them with smoother segments, allowing the flow to continue.

Key Contributions

1. Convergence and Error Estimation: The paper extends the convergent method where error bounds are provided for approximations, indicating optimal-order $H^1$-norm errors until singularities occur. However, the analysis does not cover post-singularity convergence, a noteworthy limitation given the challenges in simulating through singularities.
2. Algorithmic Detail and Comparisons: Explored in detail, the numerical surgery algorithm outlines the procedural steps involving detecting high curvature regions, surgically altering the surface topology, and resuming the flow. Comparisons against previous methods by Dziuk and others highlight the proposed surgery approach's ability to handle topological changes more gracefully.
3. Demonstrative Experiments: Extensive numerical experiments validate the algorithm's efficacy in handling singularities and demonstrate the flow's progression through illustrative surfaces, including 'dumbbell'-shaped and torus-sphere configurations. The experiments highlight the practical capability of numerical surgery to resolve pinch-offs and evolving surface geometries effectively.

Implications and Future Directions

From a theoretical standpoint, this work provides a significant step in marrying numerical methods with analytical geometric flow approaches, allowing researchers to tackle singularities in MCF computationally. The paper suggests potential extensions of this numerical surgery methodology to other geometric flows, such as forced mean curvature flow and fourth-order flows like WiLLMore flow, promising broader applicability in simulating complex surface phenomena.

Future work could focus on improving the existing algorithm to demonstrate convergence through singularities consistently. Additionally, incorporating adaptivity in mesh size and time-stepping is essential to more accurately resolve surface features with high curvature using this numerical framework. This research progresses the field towards more robust and capable simulations of dynamic surface flows, with applications in physics, materials science, and biological systems modeling.

The paper positions itself as a substantial contribution to computational mathematics by providing a practical tool for simulating MCF with topology changes, enabling exploration of new applications in various scientific and engineering domains. The proposed methodologies and results lay the groundwork for further investigations into more efficient and accurate numerical schemes in differential geometry and geometric analysis.