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GenEO spectral coarse spaces in SPD domain decomposition

Nicole Spillane

TL;DR

The paper addresses scalable two-level domain decomposition for symmetric positive definite systems by introducing GenEO coarse spaces within the abstract Schwarz framework. Coarse spaces are constructed from per-subdomain generalized eigenproblems, enabling coarse corrections via projection and allowing explicit spectral bounds on the preconditioned operator that are independent of the number of subdomains and coefficient jumps. The authors extend the abstract Schwarz theory to handle singular local solvers and projected coarse spaces, proving upper and lower eigenvalue bounds that depend on coarse-space thresholds and the coloring constant rather than problem size. The theory is illustrated on a 2D linear elasticity problem with Additive Schwarz, Neumann–Neumann, and Inexact Schwarz preconditioners, showing robust convergence, favorable conditioning, and clear guidance on scaling and overlap choices for practical, scalable solvers.

Abstract

Two-level domain decomposition methods are preconditioned Krylov solvers. What separates one- and two-level domain decomposition methods is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that allows to define and study a large variety of two-level methods. The objective of this article is to define, in the abstract Schwarz framework, a family of coarse spaces called the GenEO coarse spaces (for Generalized Eigenvalues in the Overlaps). In detail, this work is a generalization of several methods, each of which exists for a particular choice of domain decomposition method. The article both unifies the GenEO theory and extends it to new settings. The proofs are based on an abstract Schwarz theory which now applies to coarse space corrections by projection, and has been extended to consider singular local solves. Bounds for the condition numbers of the preconditioned operators are proved that are independent of the parameters in the problem (e.g., any coefficients in an underlying PDE or the number of subdomains). The coarse spaces are computed by finding low- or high-frequency spaces of some well-chosen generalized eigenvalue problems in each subdomain. The abstract framework is illustrated by defining two-level Additive Schwarz, Neumann-Neumann and Inexact Schwarz preconditioners for a two-dimensional linear elasticity problem. Explicit theoretical bounds as well as numerical results are provided for this example.

GenEO spectral coarse spaces in SPD domain decomposition

TL;DR

The paper addresses scalable two-level domain decomposition for symmetric positive definite systems by introducing GenEO coarse spaces within the abstract Schwarz framework. Coarse spaces are constructed from per-subdomain generalized eigenproblems, enabling coarse corrections via projection and allowing explicit spectral bounds on the preconditioned operator that are independent of the number of subdomains and coefficient jumps. The authors extend the abstract Schwarz theory to handle singular local solvers and projected coarse spaces, proving upper and lower eigenvalue bounds that depend on coarse-space thresholds and the coloring constant rather than problem size. The theory is illustrated on a 2D linear elasticity problem with Additive Schwarz, Neumann–Neumann, and Inexact Schwarz preconditioners, showing robust convergence, favorable conditioning, and clear guidance on scaling and overlap choices for practical, scalable solvers.

Abstract

Two-level domain decomposition methods are preconditioned Krylov solvers. What separates one- and two-level domain decomposition methods is the presence of a coarse space in the latter. The abstract Schwarz framework is a formalism that allows to define and study a large variety of two-level methods. The objective of this article is to define, in the abstract Schwarz framework, a family of coarse spaces called the GenEO coarse spaces (for Generalized Eigenvalues in the Overlaps). In detail, this work is a generalization of several methods, each of which exists for a particular choice of domain decomposition method. The article both unifies the GenEO theory and extends it to new settings. The proofs are based on an abstract Schwarz theory which now applies to coarse space corrections by projection, and has been extended to consider singular local solves. Bounds for the condition numbers of the preconditioned operators are proved that are independent of the parameters in the problem (e.g., any coefficients in an underlying PDE or the number of subdomains). The coarse spaces are computed by finding low- or high-frequency spaces of some well-chosen generalized eigenvalue problems in each subdomain. The abstract framework is illustrated by defining two-level Additive Schwarz, Neumann-Neumann and Inexact Schwarz preconditioners for a two-dimensional linear elasticity problem. Explicit theoretical bounds as well as numerical results are provided for this example.

Paper Structure

This paper contains 28 sections, 13 theorems, 69 equations, 4 figures, 4 tables.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Notation
  4. Assumptions, GenEO coarse spaces and spectral results
  5. Abstract Schwarz setting and preconditioners
  6. Notation
  7. GenEO coarse spaces and convergence results
  8. Introduction to the proofs
  9. Convergence proofs for the projected preconditioner
  10. Spectral bounds in the abstract framework
  11. Proof of \ref{['eq:th1-lambdamax']} for ${H} {A}{P}_0$
  12. Proof of \ref{['eq:th1-lambdamin']} for ${H} {A}{P}_0$
  13. Proof of \ref{['eq:th2-hard']}
  14. Example: 2d linear elasticity with Additive Schwarz, Neumann-Neumann and Inexact Schwarz
  15. Geometry and PDE
  16. ...and 13 more sections

Key Result

Theorem 9

For all results in the theorem, it is assumed that Assumptions ass:RsVs-Astilde-Hspd and ass:V0 hold. Let $\tau >0$ be a user-chosen threshold. (Recall Definition def:color for the coloring constant ${N^c}$ and Definition def:YLYH for the spaces $\mathcal{Y}_L$ and $\mathcal{Y}_R$ spanned by low and

Figures (4)

  • Figure 1: Partition into subdomains, distribution of E without and with harder layers.
  • Figure 2: Condition numbers for Additive Schwarz preconditioners: additive and hybrid; $\mu$-scaling and $k$-scaling; with and without layers; $\tau \in [4;\, 10;\, 100;\, 1000]$. All condition numbers are below the theoretical bound.
  • Figure 3: For each subdomain, solution of the generalized eigenvalue problem for computing $V_{0}$ in the case 'with layers'. Left: $\mu$-scaling, right: $k$-scaling.
  • Figure 4: Efficiency of all methods (Condition number versus coarse space dimension). Top: the test case without layers in $E$. Bottom: test case with layers in $E$.

Theorems & Definitions (35)

  • Definition 2: Coloring constant
  • Definition 6: Abstract Schwarz preconditioners
  • Definition 7: $\mathcal{Y}_L(\tau, {M}_{A}, {M}_{B})$ and $\mathcal{Y}_H(\tau, {M}_{A}, {M}_{B})$
  • Definition 8: Constant $C_\sharp$
  • Theorem 9: Spectral results for the projected and hybrid preconditioners
  • proof
  • Theorem 10: Spectral results for the additive preconditioner
  • proof
  • Lemma 11
  • proof
  • ...and 25 more