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Neumann-Neumann type domain decomposition of elliptic problems on metric graphs

Mihály Kovács, Mihály András Vághy

TL;DR

The paper develops a Neumann-Neumann type nonoverlapping domain decomposition for elliptic problems on metric graphs, proving that the discrete NW iteration converges to the FEM solution at a rate independent of mesh size by embedding the method in an abstract additive Schwarz framework. It shows the Schur complement’s condition number is also mesh-size independent and provides a practical implementation that competes with standard preconditioners. The approach is demonstrated on scale-free graph models (Dorogovtsev-Goltsev-Mendes and Barabási-Albert), where Neumann-Neumann preconditioning yields favorable iteration counts and runtimes for large graphs, confirming its potential for scalable PDE solvers on networks. Overall, the work establishes both theoretical guarantees and empirical efficacy for eigenvalue-robust, graph-structured domain decomposition in elliptic graph problems.

Abstract

In this paper we develop a Neumann-Neumann type domain decomposition method for elliptic problems on metric graphs. We describe the iteration in the continuous and discrete setting and rewrite the latter as a preconditioner for the Schur complement system. Then we formulate the discrete iteration as an abstract additive Schwarz iteration and prove that it convergences to the finite element solution with a rate that is independent of the finite element mesh size. We show that the condition number of the Schur complement is also independent of the finite element mesh size. We provide an implementation and test it on various examples of interest and compare it to other preconditioners.

Neumann-Neumann type domain decomposition of elliptic problems on metric graphs

TL;DR

The paper develops a Neumann-Neumann type nonoverlapping domain decomposition for elliptic problems on metric graphs, proving that the discrete NW iteration converges to the FEM solution at a rate independent of mesh size by embedding the method in an abstract additive Schwarz framework. It shows the Schur complement’s condition number is also mesh-size independent and provides a practical implementation that competes with standard preconditioners. The approach is demonstrated on scale-free graph models (Dorogovtsev-Goltsev-Mendes and Barabási-Albert), where Neumann-Neumann preconditioning yields favorable iteration counts and runtimes for large graphs, confirming its potential for scalable PDE solvers on networks. Overall, the work establishes both theoretical guarantees and empirical efficacy for eigenvalue-robust, graph-structured domain decomposition in elliptic graph problems.

Abstract

In this paper we develop a Neumann-Neumann type domain decomposition method for elliptic problems on metric graphs. We describe the iteration in the continuous and discrete setting and rewrite the latter as a preconditioner for the Schur complement system. Then we formulate the discrete iteration as an abstract additive Schwarz iteration and prove that it convergences to the finite element solution with a rate that is independent of the finite element mesh size. We show that the condition number of the Schur complement is also independent of the finite element mesh size. We provide an implementation and test it on various examples of interest and compare it to other preconditioners.
Paper Structure (14 sections, 7 theorems, 86 equations, 1 figure)

This paper contains 14 sections, 7 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The abstract problem
  4. Weak formulation and FEM
  5. Abstract additive Schwarz framework
  6. Neumann-Neumann method
  7. Continuous version
  8. Discrete version
  9. Subassembly and Schur complement systems
  10. Discrete harmonic functions
  11. Schwarz iteration
  12. Numerical experiments
  13. Dorogovtsev-Goltsev-Mendes graphs
  14. Barabási-Albert model

Key Result

Theorem 2.1

Assume that Then $T$ is invertible and where $\rho(\mathcal{E})$ is the spectral radius of the matrix $\mathcal{E}=\qty{\varepsilon_{ij}}_{i,j=1}^N$.

Figures (1)

  • Figure 1: The graphs $\mathsf{DGM}(1)$, $\mathsf{DGM}(2)$ and $\mathsf{DGM}(3)$.

Theorems & Definitions (13)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Proposition 3.5
  • proof
  • ...and 3 more