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Approximation and application of minimizing movements for surface PDE

Elliott Ginder, Karel Svadlenka, Takuma Muramatsu

TL;DR

This work develops a framework to approximate surface PDEs and constrained interfacial motions by marrying the Closest Point Method (CPM) with minimizing movements (MM). It extends the MBO and HMBO threshold-dynamics to curved surfaces, enabling surface mean curvature flow and hyperbolic mean curvature flow under area-preserving and multiphase constraints via a surface signed distance vector field. Numerical analyses for surface heat and wave equations demonstrate convergence with mesh refinement, validating the accuracy of the CPM+MM approach. The authors implement surface MBO and HMBO algorithms for two-phase and multiphase cases, including area-preserving variants, and show that area constraints can be controlled by a penalty parameter, providing a practical method for robust, geometry-aware interfacial dynamics on curved surfaces.

Abstract

By employing the closest point method, we extend the applicability of minimizing movements to the surface PDE setting. The corresponding approximation methods are created, and their convergence is observed. The numerical methods are then used to develop threshold dynamics algorithms for surface-constrained interfacial motions. In particular, show how the minimizing movements enable one to approximate multiphase, volume-preserving, curvature flows on surfaces via generalized MBO and HMBO algorithms.

Approximation and application of minimizing movements for surface PDE

TL;DR

This work develops a framework to approximate surface PDEs and constrained interfacial motions by marrying the Closest Point Method (CPM) with minimizing movements (MM). It extends the MBO and HMBO threshold-dynamics to curved surfaces, enabling surface mean curvature flow and hyperbolic mean curvature flow under area-preserving and multiphase constraints via a surface signed distance vector field. Numerical analyses for surface heat and wave equations demonstrate convergence with mesh refinement, validating the accuracy of the CPM+MM approach. The authors implement surface MBO and HMBO algorithms for two-phase and multiphase cases, including area-preserving variants, and show that area constraints can be controlled by a penalty parameter, providing a practical method for robust, geometry-aware interfacial dynamics on curved surfaces.

Abstract

By employing the closest point method, we extend the applicability of minimizing movements to the surface PDE setting. The corresponding approximation methods are created, and their convergence is observed. The numerical methods are then used to develop threshold dynamics algorithms for surface-constrained interfacial motions. In particular, show how the minimizing movements enable one to approximate multiphase, volume-preserving, curvature flows on surfaces via generalized MBO and HMBO algorithms.
Paper Structure (40 sections, 2 theorems, 68 equations, 24 figures, 5 tables)

This paper contains 40 sections, 2 theorems, 68 equations, 24 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Objectives
  4. Closest Point Method
  5. Minimizing movements
  6. Numerical calculation of PDEs on surfaces
  7. Approximation of PDEs on surfaces by CPM.
  8. Combination of CPM and MM
  9. Computational methods for the heat and wave equations on surfaces
  10. Numerical error analysis of MM for the heat equation on a surface
  11. MM and and surface heat equation: initial condition 1
  12. MM and and surface heat equation: initial condition 2
  13. MM numerical error analysis results (heat equation on the unit sphere)
  14. Numerical error Analysis for the surface wave equation
  15. Surface wave equation: initial condition 1
  16. ...and 25 more sections

Key Result

Theorem 1

Let $S \subset \mathbb{R}^3$ be a smooth surface. Let $u: \mathbb{R}^3 \rightarrow \mathbb{R}$ be an arbitrary smooth function that has a constant value in the normal direction of the surface $S$ near the surface. Then, on the surface $S$, holds cpm. Here, $\nabla_S$ designates the surface gradient on the surface $S$.

Figures (24)

  • Figure 1: Example of the closest point. ( $\boldsymbol{p}=C_S(\boldsymbol{x}),\quad\boldsymbol{x}\in\mathbb{R}^3$ )
  • Figure 8:
  • Figure 9:
  • Figure 11: Initial condition and computation result (Initial condition 1): (\ref{['heatcond1_ini']}) shows the initial condition, and the subsequent subfigures show the time evolution, in alphabetical order.
  • Figure 18: Initial condition and computation result (Initial condition 2): (\ref{['heatcond2_ini']}) shows the initial condition, and the subsequent subfigures show the time evolution, in alphabetical order.
  • ...and 19 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2