Patrick Farrell, Professor, University of Oxford

Chensong Zhang, Professor & Xin Liu, Professor

Reynolds-robust multilevel solvers for incompressible flow problems

14:20-16:00 October 29 (Tuesday), N202

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.

Bio: Professor Patrick E. Farrell is a leading scientist in the field of numerical analysis, specializing in the numerical solution of partial differential equations (PDEs) with applications across physics and chemistry. His research focuses on the development of advanced numerical algorithms, structure-preserving finite element methods, and bifurcation analysis, which are applied to problems in renewable energy, quantum mechanics, cardiac electrophysiology, and more. Currently a Full Professor at the Mathematical Institute of the University of Oxford, Professor Farrell has held various prestigious positions and received numerous accolades, including the Whitehead Prize from the London Mathematical Society. His work is influential in advancing computational methods and numerical software that address complex scientific challenges.