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A preconditioner for the grad-div stabilized equal-order finite elements discretizations of the Oseen problem

Yunhui He, Maxim Olshanskii

Abstract

The paper considers grad-div stabilized equal-order finite elements (FE) methods for the linearized Navier-Stokes equations. A block triangular preconditioner for the resulting system of algebraic equations is proposed which is closely related to the Augmented Lagrangian (AL) preconditioner. A field-of-values analysis of a preconditioned Krylov subspace method shows convergence bounds that are independent of the mesh parameter variation. Numerical studies support the theory and demonstrate the robustness of the approach also with respect to the viscosity parameter variation, as is typical for AL preconditioners when applied to inf-sup stable FE pairs. The numerical experiments also address the accuracy of grad-div stabilized equal-order FE method for the steady state Navier-Stokes equations.

A preconditioner for the grad-div stabilized equal-order finite elements discretizations of the Oseen problem

Abstract

The paper considers grad-div stabilized equal-order finite elements (FE) methods for the linearized Navier-Stokes equations. A block triangular preconditioner for the resulting system of algebraic equations is proposed which is closely related to the Augmented Lagrangian (AL) preconditioner. A field-of-values analysis of a preconditioned Krylov subspace method shows convergence bounds that are independent of the mesh parameter variation. Numerical studies support the theory and demonstrate the robustness of the approach also with respect to the viscosity parameter variation, as is typical for AL preconditioners when applied to inf-sup stable FE pairs. The numerical experiments also address the accuracy of grad-div stabilized equal-order FE method for the steady state Navier-Stokes equations.
Paper Structure (9 sections, 6 theorems, 79 equations, 5 figures, 6 tables)

This paper contains 9 sections, 6 theorems, 79 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Finite element formulation
  3. Algebraic problem and Schur complement preconditioner
  4. One eigenvalue problem
  5. Block preconditioner and convergence analysis
  6. Numerical experiments
  7. Driven cavity problem
  8. Backward facing step problem
  9. Conclusions

Key Result

Lemma 2.1

\newlabelthm:conti-stabi-conditions0 The bilinear form $\mathcal{L}(x;y)$ satisfies the following continuity and stability property: and with some positive mesh-independent constants

Figures (5)

  • Figure 1: $Q_1-Q_1$ FE solutions with and without grad-div stabilization for $Re=1000$ with $h=1/64$ and $Re=5000$ with $h=1/128$. Left panel: $u_1$-component of velocity along the vertical center line of the cavity; Right panel: $u_2$-component of velocity along the horizontal center line of the cavity. The reference data is from ghia1982high.
  • Figure 2: $Q_2-Q_2$ FE solutions with and without grad-div stabilization for $Re=1000$ and $Re=5000$ with $h=1/64$. Left panel: $u_1$-component of velocity along the vertical center line of the cavity; Right panel: $u_2$-component of velocity along the horizontal center line of the cavity. The reference data is from ghia1982high.
  • Figure 3: Exponentially spaced streamlines of $Q_2-Q_2$ FE solutions without and with grad-div stabilization with $h=1/64$ for $Re=5000$.
  • Figure 4: Flow over the backward-facing step for $Re=150$. Left panel: $Q_2$ solution with $\gamma=0$, $h=1/32$. Right panel: $Q_2$ solution with $\gamma=0.1$, $h=1/32$.
  • Figure 5: Flow over the backward-facing step for $Re=800$. Left panel: $Q_2$ solution with $\gamma=0$, $h=1/32$. Right panel: $Q_2$ solution with $\gamma=0.1$, $h=1/32$.

Theorems & Definitions (11)

  • Lemma 2.1
  • Proof 1
  • Theorem 3.1
  • Theorem 3.2
  • Proof 2
  • Lemma 4.1: starke1997field
  • Lemma 4.2
  • Proof 3
  • Theorem 4.3
  • Proof 4
  • ...and 1 more