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Novel Approach for solving the discrete Stokes problems based on Augmented Lagrangian and Global Techniques: Application to Saddle-Point Linear Systems from Incompressible flow

A. Badahmane, A. Ratnani, H. Sadok

TL;DR

The paper tackles efficient solution of large saddle-point systems from Stokes discretizations, where a velocity component splitting induces a $3\times3$ block structure. It introduces a novel augmented Lagrangian preconditioner based on global Krylov methods within a $3\times3$ framework (and compares to a $2\times2$ baseline), with inner solves performed by PCG and both standard and globalized variants analyzed. A spectral analysis provides eigenvalue bounds for the preconditioned operator, and extensive 2D numerical experiments demonstrate that the $3\times3$ strategy with $\mathcal{P}_{\gamma,\alpha,x,G}$ and $\mathcal{P}_{\gamma,\alpha,y,G}$ yields faster convergence and reduced CPU time, especially for larger problems. The work suggests a robust, scalable solver for incompressible flow discretizations, with potential extensions to multilevel preconditioners and parameter optimization for further performance gains.

Abstract

In this paper, a novel augmented Lagrangian preconditioner based on global Arnoldi for accelerating the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure, these systems typically arise from discretizing the Stokes equations using mixed-finite element methods. In practice, the components of velocity are always approximated using a single finite element space. More precisely, in two dimensions, our new approach based on standard space of scalar finite element basis functions to discretize the velocity space. This componentwise splitting can be shown to induce a natural block three-by-three structure. Spectral analyses is established for the exact versions of these preconditioners. Finally, the obtained numerical results claim that our novel approach is more efficient and robust for solving the discrete Stokes problems. The efficiency of our new approach is evaluated by measuring computational time.

Novel Approach for solving the discrete Stokes problems based on Augmented Lagrangian and Global Techniques: Application to Saddle-Point Linear Systems from Incompressible flow

TL;DR

The paper tackles efficient solution of large saddle-point systems from Stokes discretizations, where a velocity component splitting induces a block structure. It introduces a novel augmented Lagrangian preconditioner based on global Krylov methods within a framework (and compares to a baseline), with inner solves performed by PCG and both standard and globalized variants analyzed. A spectral analysis provides eigenvalue bounds for the preconditioned operator, and extensive 2D numerical experiments demonstrate that the strategy with and yields faster convergence and reduced CPU time, especially for larger problems. The work suggests a robust, scalable solver for incompressible flow discretizations, with potential extensions to multilevel preconditioners and parameter optimization for further performance gains.

Abstract

In this paper, a novel augmented Lagrangian preconditioner based on global Arnoldi for accelerating the convergence of Krylov subspace methods applied to linear systems of equations with a block three-by-three structure, these systems typically arise from discretizing the Stokes equations using mixed-finite element methods. In practice, the components of velocity are always approximated using a single finite element space. More precisely, in two dimensions, our new approach based on standard space of scalar finite element basis functions to discretize the velocity space. This componentwise splitting can be shown to induce a natural block three-by-three structure. Spectral analyses is established for the exact versions of these preconditioners. Finally, the obtained numerical results claim that our novel approach is more efficient and robust for solving the discrete Stokes problems. The efficiency of our new approach is evaluated by measuring computational time.
Paper Structure (12 sections, 1 theorem, 37 equations, 2 figures, 16 tables, 5 algorithms)

This paper contains 12 sections, 1 theorem, 37 equations, 2 figures, 16 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. The Problem Setting
  3. $3\times3$ Strategy for solving three-by-three linear system (\ref{['saddle']})
  4. Novel Augmented Lagrangian-based preconditioning and global techniques:
  5. Preconditioning:
  6. Algorithmic implementation of the augmented Lagrangian preconditioners $\mathcal{P}_{\gamma, \alpha,x}$ and $\mathcal{P}_{\gamma, \alpha,y}$.
  7. $2\times2$ Strategy for solving two-by-two linear system (\ref{['saddle2']})
  8. Novel Augmented Lagrangian-based preconditioning and global techniques:
  9. Preconditioning:
  10. Spectral analysis
  11. Numerical results
  12. Conclusion

Key Result

Theorem 3.1

Let the preconditioner $\mathcal{P}_{\gamma,\alpha}$ be defined as in (Pgamma). Then the eigenvalues of $\mathcal{P}_{\gamma,\alpha}^{-1}\bar{\mathcal{A}}_{2\times 2}$ are all real, positive and bounded. Furthermore the matrix $\mathcal{P}_{\gamma,\alpha}^{-1}\bar{\mathcal{A}}_{2\times 2}$ is diagon

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Example 1
  • Example 2