When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.

Achieving this for the stationary Navier–Stokes equations has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased.

Building on the work of Schöberl, Olshanskii, and Benzi, in this talk we present the first preconditioner for the Newton linearisation of the stationary incompressible Navier–Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The exact details of the preconditioner varies with discretisation, but the main idea is to combine augmented Lagrangian stabilisation, a custom multigrid prolongation operator involving local solves on coarse cells, and an additive patchwise relaxation on each level that captures the kernel of the divergence operator.

We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds number 10 to 5000. We also present recent extensions to apply these ideas to build parameter-robust solvers for the stationary incompressible resistive equations of magnetohydrodynamics.

This is joint work with Fabian Laakmann (Oxford) and Lawrence Mitchell (NVIDIA). Supported by the EPSRC Centre for Doctoral Training in Partial Differential Equations [grant EP/L015811/1], and by EPSRC grants EP/R029423/1 and EP/W026163/1.


[1]    J. Schöberl. Robust Multigrid Methods for Parameter Dependent Problems. PhD thesis, Johannes Kepler Universität Linz, Linz, Austria (1999).

[2]    M. Benzi and M. A. Olshanskii. An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28:2095–2113 (2006).

[3]    H. Elman, D. Silvester, and A. Wathen. Finite Elements and Fast Iterative Solvers. Oxford University Press, 2014.

University of Oxford