Robust multigrid techniques for augmented Lagrangian preconditioning of incompressible Stokes equations with extreme viscosity variations

TL;DR

This paper develops robust multigrid preconditioners for solving incompressible Stokes equations with highly variable viscosity, ensuring efficiency and scalability in large-scale simulations.

Contribution

It introduces augmented Lagrangian Schur complement preconditioners combined with multigrid methods tailored for extreme viscosity variations in Stokes problems.

Findings

The proposed methods are robust across viscosity contrasts up to ten orders of magnitude.

Numerical experiments demonstrate scalability to over 1.6 billion unknowns.

Eigenvalue estimates support the effectiveness of the Schur complement approximation.

Abstract

We present augmented Lagrangian Schur complement preconditioners and robust multigrid methods for incompressible Stokes problems with extreme viscosity variations. Such Stokes systems arise, for instance, upon linearization of nonlinear viscous flow problems, and they can have severely inhomogeneous and anisotropic coefficients. Using an augmented Lagrangian formulation for the incompressibility constraint makes the Schur complement easier to approximate, but results in a nearly singular (1,1)-block in the Stokes system. We present eigenvalue estimates for the quality of the Schur complement approximation. To cope with the near-singularity of the (1,1)-block, we extend a multigrid scheme with a discretization-dependent smoother and transfer operators from triangular/tetrahedral to the quadrilateral/hexahedral finite element discretizations $[\mathbb{Q}_k]^d\times \mathbb{P}_{k-1}^{\text{disc}}$, $k\geq 2$, $d=2,3$. Using numerical examples with scalar and with anisotropic fourth-order tensor viscosity arising from linearization of a viscoplastic constitutive relation, we confirm the robustness of the multigrid scheme and the overall efficiency of the solver. We present scalability results using up to 28,672 parallel tasks for problems with up to 1.6 billion unknowns and a viscosity contrast up to ten orders of magnitude.