Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scalable augmented Lagrangian preconditioners for fictitious domain problems

Michele Benzi, Marco Feder, Luca Heltai, Federica Mugnaioni

Abstract

We present preconditioning techniques to solve linear systems of equations with a block two-by-two and three-by-three structure arising from finite element discretizations of the fictitious domain method with Lagrange multipliers. In particular, we propose two augmented Lagrangian-based preconditioners to accelerate the convergence of iterative solvers for such classes of linear systems. We consider two relevant examples to illustrate the performance of these preconditioners when used in conjunction with flexible GMRES: the Poisson and the Stokes fictitious domain problems. A spectral analysis is established for both exact and inexact versions of the preconditioners. We show the effectiveness of the proposed approach and the robustness of our preconditioning strategy through extensive numerical tests in both two and three dimensions.

Scalable augmented Lagrangian preconditioners for fictitious domain problems

Abstract

We present preconditioning techniques to solve linear systems of equations with a block two-by-two and three-by-three structure arising from finite element discretizations of the fictitious domain method with Lagrange multipliers. In particular, we propose two augmented Lagrangian-based preconditioners to accelerate the convergence of iterative solvers for such classes of linear systems. We consider two relevant examples to illustrate the performance of these preconditioners when used in conjunction with flexible GMRES: the Poisson and the Stokes fictitious domain problems. A spectral analysis is established for both exact and inexact versions of the preconditioners. We show the effectiveness of the proposed approach and the robustness of our preconditioning strategy through extensive numerical tests in both two and three dimensions.

Paper Structure

This paper contains 21 sections, 16 theorems, 121 equations, 15 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Poisson fictitious domain problem and notation
  3. Finite element discretization and mesh assumptions
  4. Augmented Lagrangian preconditioner
  5. Spectrum of the preconditioned matrix
  6. Stokes fictitious domain problem
  7. Finite element discretization
  8. Augmented Lagrangian preconditioner
  9. Spectral analysis
  10. Spectrum of preconditioned matrix
  11. Mesh-independence for lower bound
  12. Spectral analysis of the inexact variant of $\mathcal{P}_{\gamma \delta}$
  13. Spectrum of the matrix preconditioned with the inexact variant of $\mathcal{P}_{\gamma \delta}$
  14. Numerical experiments
  15. Poisson problem
  16. ...and 6 more sections

Key Result

Proposition 1

Let $V_h$ and $\Lambda_h$ be defined as above. If $h_{\Omega}/h_{\Gamma}$ is sufficiently small and the mesh $\Gamma_h$ is quasi-uniform, then there exists $\beta_2$ independent of the mesh sizes $h_\Omega$ and $h_\Gamma$ such that the following discrete inf-sup condition holds:

Figures (15)

  • Figure 1: Model problem setting, with immersed domain $\omega$, immersed boundary $\Gamma$, and background domain $\Omega$.
  • Figure 2: Spectrum of the original matrix $\mathcal{A_\gamma}$ (top row) and $\mathcal{P}_{\gamma}^{-1} \mathcal{A_\gamma}$ (bottom row) for increasing values of $\gamma$.
  • Figure 3: Comparison of resulting sparsity patterns after different augmentations. Here we considered $\mathsf{A} \in \mathbb{R}^{578 \times 578}$, $\mathsf{B} \in \mathbb{R}^{81 \times 578}$, and $\mathsf{C} \in \mathbb{R}^{34\times 578}$. The resulting matrices stem from the discretization of the Stokes fictitious domain problem, using the configuration in Section \ref{['subsec:spectrum']}. The matrix $\mathsf{A_{\text{GD}}}$ denotes the matrix $\mathsf{A}$ augmented with the Grad-Div stabilization.
  • Figure 4: Spectrum of the original system matrix $\mathcal{A}_{\gamma\delta}$ (top row) and $\mathcal{P}^{-1}_{ \gamma \delta} \mathcal{A_{\gamma\delta}}$ (bottom row) for increasing values of $\gamma$ and $\delta$ applied to the Stokes test case.
  • Figure 5: Numerical check of the bounds for the preconditioned matrix when an inexact variant of the preconditioner is employed.
  • ...and 10 more figures

Theorems & Definitions (30)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Proposition 2
  • Proposition 3: BoffiGastaldiDLM, Proposition 13
  • proof
  • Theorem 2: CombinedInfSups, Theorem 3.1
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 20 more