Adaptive Stokes Preconditioning for Steady Incompressible Flows (Communications in Computational Physics, 2018)
This paper describes an adaptive preconditioner for numerical continuationof incompressible Navier–Stokes flows based on Stokes preconditioning [42] whichhas been used successfully in studies of pattern formation in convection. The preconditionertakes the form of the Helmholtz operator I−∆tL which maps the identity(no preconditioner) for ∆t ≪ 1 to Laplacian preconditioning for ∆t ≫ 1. It is built ona first order Euler time-discretization scheme and is part of the family of matrix-freemethods. The preconditioner is tested on two fluid configurations: three-dimensionaldoubly diffusive convection and a two-dimensional projection of a shear flow. In theformer case, it is found that Stokes preconditioning is more efficient for ∆t = $\mathcal{O}$(1),away from the values used in the literature. In the latter case, the simple use of thepreconditioner is not sufficient and it is necessary to split the system of equations intotwo subsystems which are solved simultaneously using two different preconditioners,one of which is parameter dependent. Due to the nature of these applications and theflexibility of the approach described, this preconditioner is expected to help in a widerange of applications.