Spectral Prescribed Mean Curvature

2024/06/08 by Jonas Haug, Haug, Jonas, R A Jewell +3
Physics and Astronomy · Mathematics · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2406.05566

Abstract

We consider prescribed mean curvature equations whose solutions are minimal surfaces, constant mean curvature surfaces, or capillary surfaces. We consider both Dirichlet boundary conditions for Plateau problems and nonlinear Neumann boundary conditions for capillary problems and we consider domains in R² to be rectangles, disks, or annuli. We present spectral methods for approximating solutions of the associated boundary value problems. These are either based on Chebyshev or Chebyshev-Fourier methods depending on the geometry of the domain. The non-linearity in the prescribed mean curvature equations is treated with a Newton method. The algorithms are designed to be adaptive; if the prescribed tolerances are not met then the resolution of the solution is increased until the tolerances are achieved. 22

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