Robust Multigrid Techniques for Augmented Lagrangian Preconditioning of Incompressible Stokes Equations with Extreme Viscosity Variations

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