Spectral Prescribed Mean Curvature

TL;DR

This paper develops spectral numerical methods using Chebyshev and Fourier techniques to solve prescribed mean curvature equations with various boundary conditions on different domains, employing adaptive Newton iterations for accuracy.

Contribution

It introduces spectral algorithms tailored for prescribed mean curvature problems with adaptive resolution and Newton-based nonlinearity handling.

Findings

Spectral methods effectively approximate solutions to prescribed mean curvature equations.

Adaptive resolution improves solution accuracy and convergence.

Algorithms handle complex boundary conditions on various geometries.

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 $\mathbf{R}^2$ 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