A numerical domain decomposition method for solving elliptic equations on manifolds

TL;DR

This paper introduces a novel numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds, avoiding the need for global triangulations and demonstrating effectiveness on several 4-dimensional manifolds.

Contribution

The paper presents a new domain decomposition approach that simplifies solving elliptic equations on manifolds without global meshes, which is a significant advancement.

Findings

Successfully tested on 4-dimensional manifolds such as $S^{4}$, $ ext{CP}^{2}$, and $S^{2} imes S^{2}$

Avoids the complexity of global triangulations or grids on manifolds

Demonstrates numerical effectiveness on various complex manifolds

Abstract

A new numerical domain decomposition method is proposed for solving elliptic equations on compact Riemannian manifolds. The advantage of this method is to avoid global triangulations or grids on manifolds. Our method is numerically tested on some $4$-dimensional manifolds such as the unit sphere $S^{4}$, the complex projective space $\mathbb{CP}^{2}$ and the product manifold $S^{2} \times S^{2}$.