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A numerical domain decomposition method for solving elliptic equations on manifolds

Shuhao Cao, Lizhen Qin

TL;DR

The paper develops a flexible numerical framework for elliptic PDEs on compact manifolds that avoids global triangulations by using overlapping domain decomposition with local coordinate charts. Building on Lions' Schwarz-type method, it couples subdomain solves through coordinate transitions and interpolation on nonmatching grids, and discretizes each chart with tensor-product finite elements on $d$-rectangles. The approach is validated on four-dimensional manifolds ($S^{4}$, $\mathbb{CP}^{2}$, $S^{2}\times S^{2}$), showing geometric convergence in the continuous setting and robust, convergent behavior in the discrete scheme, with iteration counts decreasing as overlap increases. This method extends DDM techniques to general high-dimensional manifolds, enabling scalable solutions without requiring a global mesh.

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 $S^{2} \times S^{2}$.

A numerical domain decomposition method for solving elliptic equations on manifolds

TL;DR

The paper develops a flexible numerical framework for elliptic PDEs on compact manifolds that avoids global triangulations by using overlapping domain decomposition with local coordinate charts. Building on Lions' Schwarz-type method, it couples subdomain solves through coordinate transitions and interpolation on nonmatching grids, and discretizes each chart with tensor-product finite elements on -rectangles. The approach is validated on four-dimensional manifolds (, , ), showing geometric convergence in the continuous setting and robust, convergent behavior in the discrete scheme, with iteration counts decreasing as overlap increases. This method extends DDM techniques to general high-dimensional manifolds, enabling scalable solutions without requiring a global mesh.

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 -dimensional manifolds such as the unit sphere , the complex projective space and the product manifold .
Paper Structure (10 sections, 2 theorems, 89 equations, 2 figures, 16 tables, 2 algorithms)

This paper contains 10 sections, 2 theorems, 89 equations, 2 figures, 16 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theory on Continuous Problems
  3. Numerical Scheme
  4. Finite Element Spaces over a $d$-Rectangle
  5. Discrete Iterative Procedure
  6. Product Manifolds
  7. Numerical Experiments
  8. Two Problems on S4
  9. A Problem on CP2
  10. A Problem on S2xS2

Key Result

Theorem 2.1

\newlabelthm_convergence0 There exist constants $C_{0} >0$ and $L \in [0,1)$ such that, $\forall u^{0} \in H^{1}_{0} (M)$, $\forall n>0$, where $u$ is the solution to eqn_problem_weak and $u^{n}$ is the $n$th iterated approximation in Algorithm alg_continuous with initial guess $u^{0}$.

Figures (2)

  • Figure 1: An illustration of a transition of coordinates on $S^m$.
  • Figure 1: An illustration of the stereographic projections used in the case of $S^4$.

Theorems & Definitions (5)

  • Theorem 2.1
  • Lemma 2.2
  • Proof 1: Proof of Theorem \ref{['thm_convergence']}
  • Remark 3.1
  • Remark 5.1