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New preconditioner strategy for solving block four-by-four linear systems: An application to the saddle-point problem from 3D Stokes equation

Achraf Badahmane, Ahmed Ratnani, Hassane Sadok

Abstract

We have presented a fast method for solving a specific type of block four-by-four saddlepoint problem arising from the finite element discretization of the generalized 3D Stokes problem. We analyze the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Furthermore, we suggested utilizing the preconditioned global conjugate gradient method (PGCG) as a block iterative solver for handling multiple right-hand sides within the sub-system and give some new convergence results. Numerical experiments have shown that our preconditioned iterative approach is very efficient for solving the 3D Stokes problem

New preconditioner strategy for solving block four-by-four linear systems: An application to the saddle-point problem from 3D Stokes equation

Abstract

We have presented a fast method for solving a specific type of block four-by-four saddlepoint problem arising from the finite element discretization of the generalized 3D Stokes problem. We analyze the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Furthermore, we suggested utilizing the preconditioned global conjugate gradient method (PGCG) as a block iterative solver for handling multiple right-hand sides within the sub-system and give some new convergence results. Numerical experiments have shown that our preconditioned iterative approach is very efficient for solving the 3D Stokes problem
Paper Structure (11 sections, 5 theorems, 70 equations, 1 figure, 6 tables, 2 algorithms)

This paper contains 11 sections, 5 theorems, 70 equations, 1 figure, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setting and variational formulations
  3. Finite element discretization with $P_{1}$-Bubble/$P_{1}$
  4. Preconditioned Krylov subspace methods
  5. Global regularized preconditioner
  6. Properties of the preconditioner $\mathcal{P}_{r}$
  7. Implementation of the regularized preconditioner $\mathcal{P}_{r}$ (\ref{['Pr']}).
  8. Global Krylov subspace methods
  9. Convergence result for the preconditioned global conjugate gradient method ($\mathcal{P}$GCG)
  10. Numerical results
  11. Conclusion

Key Result

Proposition 1

badahmane The regularized preconditioner (Pr) has the block-triangular factorization: where $\mathcal{S}=\beta Q-B(A^{-1}\otimes I_3)B^{T}$. If $\beta \lambda_{\min}(Q) > \lambda_{\max}(B(A^{-1}\otimes I_3)B^T)$, then $\mathcal{S}$ is a positive definite matrix.

Figures (1)

  • Figure 1:

Theorems & Definitions (14)

  • Proposition 1
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • Definition 3.3
  • Definition 3.4
  • Theorem 3.1
  • Theorem 3.2
  • ...and 4 more