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

TL;DR

This paper introduces a new block triangular preconditioner for grad-div stabilized equal-order finite element discretizations of the Oseen problem, improving solver robustness and efficiency across mesh and viscosity variations.

Contribution

It proposes a preconditioner related to the Augmented Lagrangian approach, with a field-of-values analysis showing mesh-independent convergence bounds.

Findings

Preconditioner demonstrates mesh-independent convergence.

Numerical results show robustness against viscosity variations.

The approach improves solver efficiency for steady Navier-Stokes equations.

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.