We describe a multiplicative low-rank correction scheme for pressure Schur complement preconditioners to accelerate the iterative solution of the linearized Navier-Stokes equations. The application of interest is a model for buoyancydriven fluid flows described by the Boussinesq approximation which combines the Navier-Stokes equations enhanced with a Coriolis term and a temperature advection-diffusion equation. The update method is based on a low-rank approximation to the error between the identity and the preconditioned Schur complement. Numerical results on a cube and a shell geometry illustrate the action of the lowrank correction on spectra of preconditioned Schur complements using known preconditioning techniques, the least-squares commutator and the SIMPLE method. The computational costs of the update method are also investigated. The goal is to analyze whether such an update method can lead to accelerated solvers. Numerical experiments show that the update technique can reduce iteration counts in some cases but (counter-intutively) may increase iteration counts in other settings.

This contribution discusses the construction and the utility of anisotropic kernels for numerical fluid flow simulation. So far, commonly used radial kernels, such as Gaussians, (inverse) multiquadrics and polyharmonic splines, were proven to be powerful tools in various applications of multivariate scattered data approximation. Due to the well-known uncertainty principle, however, their resulting reconstruction methods are often critical when it comes to combine high order approximation with numerical stability. In many cases this leads to severe limitations, especially when it comes to fluid flow simulations. Therefore, more sophisticated kernel methods are required. In this paper, we show how to obtain anisotropic positive definite kernels from standard kernels rather directly. Our proposed construction yields a new class of more flexible kernels that are particularly useful for fluid flow simulations. To this end, the finite volume particle method is used as a prototype of our discussion, where scattered data approximation is needed in the recovery step of weighted essentially non-oscillatory (WENO) reconstructions. Supporting numerical examples and comparisons are provided.