A convergent finite element algorithm for mean curvature flow in arbitrary codimension (2107.10577v2)

Abstract: Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulae. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical experiments in codimension 2 illustrate and complement our theoretical results.

• The paper develops a convergent finite element method to simulate mean curvature flow in arbitrary codimensions with optimal error control.
• It employs evolving surface finite elements and linearly implicit BDF schemes for robust and stable numerical simulation.
• Numerical experiments in codimension 2 validate the method’s accuracy and its potential for complex geometric applications.

A Convergent Finite Element Algorithm for Mean Curvature Flow in Arbitrary Codimension

The paper, authored by Tim Binz and Balázs Kovács, introduces a novel algorithm to address the computational challenges associated with the mean curvature flow (MCF) of surfaces in arbitrary codimension. This research aims to provide a robust computational method to simulate and analyze the evolution of a closed $m$-dimensional surface under MCF, a significant area of interest in geometric analysis and computer graphics.

The primary contribution of this work lies in developing a finite element method (FEM) that remains uniform in time with respect to the $H^1$-norm. The proposed algorithm achieves optimal-order error estimates for both semi- and fully discrete versions applied to MCF processes, particularly in scenarios where codimension is two or greater.

Methodology

The research leverages advanced mathematical frameworks to derive new parabolic evolution equations specifically for the mean curvature vector $H$ and the orthogonal projection $\pi$ onto the tangent space of the evolving surface. The finite element method designed in this paper utilizes evolving surface finite elements and implements linearly implicit backward difference formulae (BDF) of orders two through five, emphasizing computational efficiency and stability under a mild step-size restriction.

Results

The authors demonstrate the feasibility and accuracy of the proposed method through a series of numerical experiments conducted in codimension 2 spaces. These simulations not only complement the theoretical findings but establish the devised algorithm's consistency with respect to traditional approaches, such as those developed for mean curvature flows in codimension one. One critical observation is the method’s capability to handle more complex nonlinear terms inherent in higher codimension flows, which are locally Lipschitz continuous and consequently support convergence alike proofs available for simpler flows.

Implications and Future Work

Binz and Kovács's findings hold significant promise for computational geometry and physical simulations involving higher codimensional surfaces. The implications of this research extend into various realms like materials science, computer-aided geometric design, and biological modeling where understanding the interface evolution is paramount.

Future research directions will likely focus on refining the proposed FEM framework for even higher dimensions and codimensions, addressing computational overheads while ensuring error minimization and stability. Moreover, the general adaptability of this method provides a foundation for potential integration with other geometric flow models, further expanding its application scope.

In sum, this paper presents a mathematically rigorous and computationally efficient approach for simulating mean curvature flow in arbitrary codimension, with a clear direction for future exploration and enhancement in related computational fields.