A convergent finite element algorithm for generalized mean curvature flows of closed surfaces (2010.11044v2)

Abstract: An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full discretisations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to non-linear second-order parabolic evolution equations for the normal velocity and normal vector. Convergence proofs are presented in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity, and therefore for the mean curvature. The stability analysis is performed in the matrix-vector formulation, and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results, and also to report on monotone quantities, e.g.~Hawking mass for inverse mean curvature flow. Complemented by experiments for non-convex surfaces.

• The paper introduces a convergent finite element algorithm combining ESFEM spatial discretization with BDF time integration to simulate generalized mean curvature flows.
• The paper establishes rigorous error estimates and optimal order convergence in the H1-norm, ensuring stability independent of classical geometric estimates.
• The method’s versatility in handling various curvature flows offers promising applications in material science and image processing.

Finite Element Algorithm for Generalized Mean Curvature Flows

This paper introduces a convergent finite element algorithm for simulating generalized mean curvature flows (MCF) of closed surfaces. Unlike the classical mean curvature flow, generalized MCF encompasses a broader class, including the inverse mean curvature flow (IMCF) and its various generalizations. The method presented is versatile enough to handle these different flows by employing evolving surface finite elements (ESFEM) for spatial discretization, coupled with linearly implicit backward difference formulae (BDF) for time integration.

Numerical Method and Convergence

The algorithm combines ESFEM and a range of BDF methods of orders two to five. It solves the coupled system of equations governing the surface position, normal vector, and the normal velocity driven by the curvature. The spatial ESFEM discretization employs finite elements of polynomial degree at least two, while full discretization employs BDF methods. The paper provides rigorous error estimates for both semi-discretizations and full discretizations, validating optimal order convergence in the $H^1$-norm. This verifies that the method achieves the expected convergence rates for the considered classes of curvature flows given sufficiently regular solutions.

Key Contributions

One significant contribution of this work is the provision of a convergence proof for these generalized curvature flows, a topic with limited results in existing literature prior to this paper. In classical numerical approaches for curvature flows, stability proofs often rely on geometric estimates. However, this approach establishes stability and consistency independently, a novel aspect in the context of discretized continuum mechanics. Stability is assessed in a matrix-vector framework, which uses energy-based arguments facilitated by techniques from Dahlquist's G-stability theory and Nevanlinna-Odeh's multiplier approach.

Implications and Future Work

From a theoretical perspective, this work enriches the understanding of generalized mean curvature flows, offering tools that can potentially adapt to more complex geometric scenarios. Practically, the method's adaptability to various forms of mean curvature flows positions it as a viable tool in fields such as material science and image processing, where surface evolution is often governed by complex curvature-dependent laws.

This methodology opens up several avenues for future research, particularly for exploring these discrete methods on even richer classes of geometric flows or addressing challenges where singularity formation becomes unavoidable—essentially situations beyond the smooth surface framework assumed here. Furthermore, the convergence and stability approaches might inspire similar techniques in other areas of computational geometry and physics where governing equations exhibit analogous nonlinear and geometric properties.