A computational approach to an optimal partition problem on surfaces

TL;DR

This paper presents a computational method for solving an optimal partition problem on surfaces, aiming to minimize eigenvalues of the Laplace-Beltrami operator, with applications demonstrated on spheres and other surfaces.

Contribution

It introduces a novel eigenfunction segregation-based computational approach and applies high performance computing to solve complex surface partition problems.

Findings

Method accurately computes partitions on the sphere with three segments.

The approach extends to higher numbers of partitions and different surfaces.

Results demonstrate the effectiveness of the computational approach.

Abstract

We explore an optimal partition problem on surfaces using a computational approach. The problem is to minimise the sum of the first Dirichlet Laplace--Beltrami operator eigenvalues over a given number of partitions of a surface. We consider a method based on eigenfunction segregation and perform calculations using modern high performance computing techniques. We first test the accuracy of the method in the case of three partitions on the sphere then explore the problem for higher numbers of partitions and on other surfaces.