Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Analysis of Overlapping Schwarz Methods for Vector Field Problems in Three Dimensions with Generally Shaped Domains

Duk-Soon Oh, Shangyou Zhang

TL;DR

This work develops a topology-robust two-level overlapping Schwarz framework for discretized vector-field problems in three dimensions, specifically targeting $H(\mathbf{curl})$ with Nédélec elements and $H(\mathrm{div})$ with Raviart-Thomas elements. The approach hinges on robust regular decompositions (Hiptmair-Pechstein) and a Schwarz framework with coarse and local corrections, yielding explicit condition-number bounds that scale linearly with the overlap parameter via $\max_i(1+ H_i/\delta_i)$; these results hold under modest assumptions (e.g., $\eta_c,\eta_d \le 1$) and can be sharpened when topological invariants such as Betti numbers vanish. The authors provide detailed stability and approximation lemmas for the cochain projections, derive the two-level preconditioners, and prove the main $H(\mathbf{curl})$ and $H(\mathrm{div})$ condition-number estimates; numerical experiments in 2D and 3D with ND/RT elements confirm optimal convergence and illustrate bounded constants even on non-convex domains, aligning with the theory. Overall, the paper delivers topology-robust, scalable solvers for Maxwell-type and related vector problems on general domains, with potential impact on eddy-current simulations and mixed formulations in fluid and solid mechanics.

Abstract

This paper introduces a novel approach to analyzing overlapping Schwarz methods for Nédélec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.

New Analysis of Overlapping Schwarz Methods for Vector Field Problems in Three Dimensions with Generally Shaped Domains

TL;DR

This work develops a topology-robust two-level overlapping Schwarz framework for discretized vector-field problems in three dimensions, specifically targeting with Nédélec elements and with Raviart-Thomas elements. The approach hinges on robust regular decompositions (Hiptmair-Pechstein) and a Schwarz framework with coarse and local corrections, yielding explicit condition-number bounds that scale linearly with the overlap parameter via ; these results hold under modest assumptions (e.g., ) and can be sharpened when topological invariants such as Betti numbers vanish. The authors provide detailed stability and approximation lemmas for the cochain projections, derive the two-level preconditioners, and prove the main and condition-number estimates; numerical experiments in 2D and 3D with ND/RT elements confirm optimal convergence and illustrate bounded constants even on non-convex domains, aligning with the theory. Overall, the paper delivers topology-robust, scalable solvers for Maxwell-type and related vector problems on general domains, with potential impact on eddy-current simulations and mixed formulations in fluid and solid mechanics.

Abstract

This paper introduces a novel approach to analyzing overlapping Schwarz methods for Nédélec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.
Paper Structure (13 sections, 23 theorems, 93 equations, 5 figures, 6 tables)

This paper contains 13 sections, 23 theorems, 93 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The discrete problems
  3. Overlapping Schwarz methods
  4. Condition number estimate
  5. Preliminaries
  6. Regular decompositions for vector fields
  7. Schwarz framework
  8. Condition number estimate for $H(\mathbf{curl})$
  9. Condition number estimate for $H(\mathrm{div})$
  10. Numerical experiments
  11. Nédélec Type-1 rectangular element
  12. Triangular Nédélec element
  13. Tetrahedral Nédélec element

Key Result

Lemma 4

For $\bm{u}, \bm{p} \in \left(H^1(\Omega)\right)^3$, the following estimates hold: with constants independent of $\bm{u}$, $\bm{p}$, and $H$.

Figures (5)

  • Figure 5.1: The first three grids for computing Tables \ref{['t1']}--\ref{['t2']}.
  • Figure 5.2: Left: The level three function nodes in the four subdomains, where $\Omega=(0,1)^2$; Right: The 4-subdomain nodes for $\Omega= (0,1)^2\setminus\{\frac{1}{2}\}\times(0,\frac{1}{2}]$.
  • Figure 5.3: The first three grids for computing Tables \ref{['t21']}--\ref{['t22']}.
  • Figure 5.4: Left: The level three function nodes in the four subdomains, where $\Omega=(0,1)^2$; Right: The 4-subdomain nodes for $\Omega= (0,1)^2\setminus\{\frac{1}{2}\}\times(0,\frac{1}{2}]$.
  • Figure 5.5: The first two grids for the computation in Tables \ref{['t31']}--\ref{['t32']}.

Theorems & Definitions (37)

  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8: Hiptmair-Pechstein decomposition for edge elements
  • Lemma 9: Hiptmair-Pechstein decomposition for face elements
  • Lemma 10
  • Lemma 11
  • Lemma 12
  • Lemma 13
  • ...and 27 more