High-order algorithms for solving eigenproblems over discrete surfaces

TL;DR

This paper introduces a new high-order, convergent algorithm combining local tangential lifting with configuration equations to accurately solve eigenproblems of Laplace-Beltrami operators on discrete surfaces, with applications in geometric analysis.

Contribution

The paper develops a novel high-order algorithm for eigenproblems on discrete surfaces, achieving improved convergence rates and enabling computation of geometric invariants.

Findings

Convergence rate of $O(r^n)$ for the proposed algorithms.

Effective computation of eigenvalues on various surfaces.

Discussion of high-order accuracy for geometric invariants.

Abstract

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting (LTL) method with the configuration equation to develop a new effective and convergent algorithm to solve the eigenvalue problems of the Laplace-Beltrami operators acting on functions over discrete surfaces. The convergence rates of our algorithms of discrete Laplace-Beltrami operators over surfaces is $O(r^n)$, $n \geq 1$, where $r$ represents the size of the mesh of discretization of the surface. The problem of high-order accuracies will also be discussed and used to compute geometric invariants of the underlying surfaces. Some convergence tests and eigenvalue computations on the sphere, tori and a dumbbell are presented.