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SUMMARY:A Reynolds-robust preconditioner for the 3D stationary Navier-Stok
 es equations - Patrick Farrell\, Mathematical Institute\, University of Ox
 ford
DTSTART:20191029T130000Z
DTEND:20191029T140000Z
UID:TALK133327@talks.cam.ac.uk
CONTACT:Chris Richardson
DESCRIPTION:When approximating PDEs with the finite element method\, large
  sparse\nlinear systems must be solved. The ideal preconditioner yields\nc
 onvergence that is algorithmically optimal and parameter robust\, i.e.\nth
 e number of Krylov iterations required to solve the linear system to a\ngi
 ven accuracy does not grow substantially as the mesh or problem\nparameter
 s are changed.\n\nAchieving this for the stationary Navier-Stokes has prov
 en challenging:\nLU factorisation is Reynolds-robust but scales poorly wit
 h degree of\nfreedom count\, while Schur complement approximations such as
  PCD and LSC\ndegrade as the Reynolds number is increased.\n\nBuilding on 
 the ideas of Schöberl\, Benzi & Olshanskii\, in this talk we\npresent the
  first preconditioner for the Newton linearisation of the\nstationary Navi
 er–Stokes equations in three dimensions that achieves\nboth optimal comp
 lexity and Reynolds-robustness. The scheme combines\naugmented Lagrangian 
 stabilisation to control the Schur complement\, the\nconvection stabilisat
 ion proposed by Burman & Hansbo\, a\ndivergence-capturing additive Schwarz
  relaxation method on each level\,\nand a specialised prolongation operato
 r involving non-overlapping local\nStokes solves. The properties of the pr
 econditioner are tailored to the\ndivergence-free Scott–Vogelius discret
 isation.\n\nWe present 3D simulations with over one billion degrees of fre
 edom with\nrobust performance from Reynolds numbers 10 to 5000.
LOCATION:JJ Thomson Seminar Room\, Maxwell Centre\, Cavendish Laboratory
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