Speaker: Professor Patrick Farrell, University of Oxford

Inviter: Chensong Zhang, Professor & Xin Liu, Professor

Title: Reynolds-robust multilevel solvers for incompressible flow problems

Language: English

Time & Venue: 2024.10.29 14:20-16:00 N202

Abstract: 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 has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom (dof) 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 Navier--Stokes equations in three dimensions that achieves both optimal complexity in dof count and Reynolds-robustness. The exact details of the preconditioner varies with discretisation, but the general theme 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 simulations with robust performance over Reynolds numbers 10 to 10000, several orders of magnitude better than what was previously possible. We also present an extension to a system of equations arising in the magnetohydrodynamics of liquid metals.