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A BFBt preconditioner for Double Saddle-Point Systems

Chen Greif

Abstract

We consider block preconditioners for double saddle-point systems, and investigate the effect of approximating the nested Schur complement associated with the trailing diagonal block on the eigenvalue distribution of the preconditioned matrix. We develop a variant of Elman's BFBt method and adapt it to this family of linear systems. Our findings are illustrated on a Marker-and-Cell discretization of the Stokes-Darcy equations.

A BFBt preconditioner for Double Saddle-Point Systems

Abstract

We consider block preconditioners for double saddle-point systems, and investigate the effect of approximating the nested Schur complement associated with the trailing diagonal block on the eigenvalue distribution of the preconditioned matrix. We develop a variant of Elman's BFBt method and adapt it to this family of linear systems. Our findings are illustrated on a Marker-and-Cell discretization of the Stokes-Darcy equations.
Paper Structure (18 sections, 1 theorem, 56 equations, 2 figures, 3 tables)

This paper contains 18 sections, 1 theorem, 56 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. An inexact $S_2$
  3. Eigenvalues of $\mathcal{\widehat{M}}_{LT}^{-1} \mathcal{K}$
  4. Eigenvalues of $\mathcal{\widehat{M}}_{D}^{-1} \mathcal{K}$
  5. $D \ne 0$ and $m \ge n$
  6. $A$ is symmetric positive definite, $D = 0$, and $n \ge m$
  7. Perturbation analysis
  8. BFBt preconditioning
  9. Application to double saddle-point systems
  10. Eigenvalue analysis
  11. A similarity transformation
  12. Additional observations for the symmetric case of Section \ref{['sec:symD0']}
  13. Application: Marker-and-Cell discretization of the Stokes--Darcy equations
  14. Validation of the eigenvalue analysis
  15. Practical approximations of the Schur complements
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose $A$ is symmetric positive definite, $D=0$, and $n \ge m$, and suppose further that $\widehat{S}_2$ is symmetric positive definite. Then, the eigenvalues of the preconditioned matrix ${{\mathcal{\widehat{M}}}_{D}^{-1}} \mathcal{K}$ defined in eq:eigMK0 are:

Figures (2)

  • Figure 1: Stokes-Darcy equations: eigenvalues of $\widehat{\mathcal{M}}_{LT}^{-1} \mathcal{K}$ in the complex plane, with $n=16$, for $\nu=\kappa=1$.
  • Figure 2: Stokes-Darcy equations: real parts of the eigenvalues of $\widehat{\mathcal{M}}_D^{-1} \mathcal{K}$, with $n=16$, for $\nu=\kappa=1$. Some of the eigenvalues are complex but their imaginary parts are smaller than $0.01$ in norm.

Theorems & Definitions (3)

  • Theorem 2.1
  • proof
  • Remark 2.2