Fast and Robust Solvers for Time-Discretised Incompressible Navier-Stokes Equations

Prof. David Silvester

University of Manchester Institute for Science and Technology (England)

In this talk we consider the design of robust and efficient methods for solving the Navier-Stokes equations governing laminar flow of a viscous incompressible fluid. Three fundamental issues will be assessed in detail: the discretisation of the convective transport terms, the (weak-) enforcement of the incompressibility constraint in a mixed finite element setting, and the solution of the indefinite (Stokes-) systems arising at each time-level. Our aim is to prescribe a framework for adaptive error control. The essential ingredients are: an unconditionally stable time-discretisation; ``natural'' spatial discretisations which are (inf-sup) stable; and a fast iterative solution strategy generating iterates converging monotonically in an appropriate norm (which mimics the dissipation inherent in the continuous system). A distinguishing feature of our methodology is the use of multigrid preconditioning to accelerate the convergence of our Krylov subspace iterative solver. We motivate this with some analysis showing that the contraction rate is bounded away from unity independently of the choice of the mixed finite element method and the subdivision parameter. Analysis and implementation of ``pure'' multigrid methods seems to be relatively complicated and (discretisation-) method dependent by comparison. This is joint work with Andy Wathen.