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SUMMARY:A Reynolds-robust preconditioner for the 3D stationary Navier-Stok
es equations - Patrick Farrell (University of Oxford)
DTSTART:20191031T160000Z
DTEND:20191031T170000Z
UID:TALK134209@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:When approximating PDEs with the finite element method\, large
sparse linear systems must be solved. The ideal preconditioner yields con
vergence 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 paramet
ers are changed. \; Achieving this for the stationary Navier-Stokes
has proven challenging: LU factorisation is Reynolds-robust but scales po
orly with degree of freedom count\, while Schur complement approximations
such as PCD and LSC degrade as the Reynolds number is increased. \;
Building on ideas of Schö\;berl\, Xu\, Zikatanov\, Benzi &\; Olsha
nskii\, in this talk we present the first preconditioner for the Newton li
nearisation of the stationary Navier&ndash\;Stokes equations in three dime
nsions that achieves both optimal complexity and Reynolds-robustness. The
scheme combines augmented Lagrangian stabilisation to control the Schur co
mplement\, the convection stabilisation proposed by Douglas &\; Dupont\
, a divergence-capturing additive Schwarz relaxation method on each level\
, and a specialised prolongation operator involving non-overlapping local
Stokes solves. The properties of the preconditioner are tailored to the di
vergence-free CG(k)-DG(k-1) discretisation and the appropriate relaxation
is derived from considerations of finite element exterior calculus.
\; We present 3D simulations with over one billion degrees of freedom wit
h robust performance from Reynolds numbers 10 to 5000.
LOCATION:Seminar Room 2\, Newton Institute
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