Numerical analysis for the Plateau problem by the method of fundamental solutions

TL;DR

This paper introduces a fast, accurate numerical method based on the fundamental solutions to identify minimal surfaces with shared boundaries, avoiding mesh division and ensuring smooth approximations.

Contribution

It develops a novel numerical scheme with convergence and error analysis for the Plateau problem using the method of fundamental solutions, enabling smooth surface approximations.

Findings

High-speed, high-accuracy numerical scheme for minimal surfaces

Convergence analysis for Dirichlet energy

Error bounds for mean curvature approximation

Abstract

Towards identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental solutions. We establish the convergence analysis for Dirichlet energy and $L^\infty$-error analysis for mean curvature. Each of the approximate solutions in our scheme is a smooth surface, which is a significant difference from previous studies that required mesh division.