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A parallel domain decomposition method for solving elliptic equations on manifolds

Lizhen Qin, Feng Wang, Yun Wang

TL;DR

The paper tackles elliptic PDEs on manifolds where global grids are impractical by introducing a parallel overlapping domain decomposition method inspired by Lions. By combining local Euclidean reductions with a partition of unity and interpolation across nonmatching grids, the approach yields a highly parallel scheme with solid convergence theory for the continuous problem and a practical, FE-based discrete scheme. The authors prove convergence results for the continuous method and validate the numerical scheme on 4D manifolds CP$^2$, $B^4$, and $B^2\times S^2$ (with and without boundary), demonstrating accurate solutions and favorable parallel performance. This work provides a scalable framework for solving Laplace-type problems on complex manifolds without requiring global triangulations, with clear applicability to high-dimensional geometric settings.

Abstract

We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds. Additionally, it features a highly parallel iterative scheme. To verify its efficacy, we conduct numerical experiments on some $4$-dimensional manifolds without and with boundary.

A parallel domain decomposition method for solving elliptic equations on manifolds

TL;DR

The paper tackles elliptic PDEs on manifolds where global grids are impractical by introducing a parallel overlapping domain decomposition method inspired by Lions. By combining local Euclidean reductions with a partition of unity and interpolation across nonmatching grids, the approach yields a highly parallel scheme with solid convergence theory for the continuous problem and a practical, FE-based discrete scheme. The authors prove convergence results for the continuous method and validate the numerical scheme on 4D manifolds CP, , and (with and without boundary), demonstrating accurate solutions and favorable parallel performance. This work provides a scalable framework for solving Laplace-type problems on complex manifolds without requiring global triangulations, with clear applicability to high-dimensional geometric settings.

Abstract

We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds. Additionally, it features a highly parallel iterative scheme. To verify its efficacy, we conduct numerical experiments on some -dimensional manifolds without and with boundary.
Paper Structure (13 sections, 7 theorems, 106 equations, 3 figures, 6 tables, 2 algorithms)

This paper contains 13 sections, 7 theorems, 106 equations, 3 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Theory on Continuous Problems
  3. Model Problem and Continuous Algorithm
  4. Subsolutions and Supersolutions
  5. Proof of Convergence
  6. Numerical Scheme
  7. Finite Element Spaces over a $d$-Rectangle
  8. Numerical Algorithm
  9. Manifolds with Boundary
  10. Numerical Experiments
  11. A Problem on CP2
  12. A Problem on B4
  13. A Problem on B2xS2

Key Result

Lemma 2.3

Under the Assumptions asp_decomposition and asp_partition, in alg_continuous_1, we have $\forall n$, $\forall i$, $u^{n} \in C^{0} (M) \cap H^{1} (M)$, $u^{n}_{i} \in C^{0} (M_{i}) \cap H^{1} (M_{i})$, $u^{n}|_{\partial M} = \varphi$, and $u^{n}_{i}|_{\partial M} = \varphi$.

Figures (3)

  • Figure 1: An illustration of domain decomposition.
  • Figure 2: An illustration of domain decomposition.
  • Figure 3: An illustration of domain decomposition.

Theorems & Definitions (17)

  • Lemma 2.3
  • Proof 1
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • Proof 2
  • Lemma 2.9
  • Proof 3
  • ...and 7 more