Obstacle Mean Curvature Flow: Efficient Approximation and Convergence Analysis

TL;DR

The paper presents an efficient obstacle-aware MBO scheme that enforces obstacle constraints at each step while preserving stability, monotonicity, and a gradient-flow interpretation. It proves convergence of the scheme to obstacle mean curvature flow in the viscosity sense and shows consistency with space-discrete approximations via Gamma-convergence. The authors develop a space-time discretization framework based on heat-kernel diffusion and thresholding, and demonstrate numerical simulations of invasion processes with complex obstacles, achieving scalable performance using FFT-based implementations. The work provides a solid theoretical foundation and practical tools for simulating curvature-driven interfaces in obstacle-rich settings, with potential applications in biology and materials science.

Abstract

We introduce a simple and efficient numerical method to compute mean curvature flow with obstacles. The method augments the Merrimam-Bence-Osher scheme with a pointwise update that enforces the constraint and therefore retains the computational complexity of the original scheme. Remarkably, this naive scheme inherits both crucial structural properties of obstacle mean curvature flow: a geometric comparison principle and a minimizing movements interpretation. The latter immediately implies the unconditional stability of the scheme. Based on the comparison principle we prove the convergence of the scheme to the viscosity solution of obstacle mean curvature flow. Moreover, using the minimizing movements interpretation, we show convergence of a spatially discrete model. Finally, we present numerical experiments for a physical model that inspired this work.

Figures (4)

• Figure 1: Schematic illustration of the construction of initial conditions $u_0, \psi, \phi$.
• Figure 2: Simulation of mean curvature flow with obstacles computed with the scheme \ref{['def:discrete_scheme']}. From left to right: initial condition, after 2000, 4000, 6000 and 8000 iterations.
• Figure 3: Comparison of average running time per iteration of our scheme described in Section \ref{['sec:numerics']}. The average was taken over $100$ iterations.
• Figure 4: Simualation to visualize numerical errors in the steady states. Fig. \ref{['subfig:a']} shows schematic the initial configuration which are at the same time the obstacles. Fig. \ref{['subfig:b']}, \ref{['subfig:c']} and \ref{['subfig:d']} show the steady states of our scheme \ref{['eq:implemented_scheme']} for different diffusion times $h$ but fixed spatial discretization $\varepsilon = 0.001$. In Fig. \ref{['subfig:b']} the diffusion time $h$ is not big enough for the given spatial resolution such that the convex hull - the steady state of the obstacle mean curvature flow - is not well approximated. In Fig. \ref{['subfig:c']}$h$ is chosen well for the example. In Fig. \ref{['subfig:d']}$h$ is too big such that non-touching areas get connected.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Remark 1
• Theorem 1: Theorem 4.2 in zbMATH06841741
• Proposition 1: Maximal subsolution, minimal supersolution
• proof
• Proposition 2
• proof
• Theorem 2
• proof