Peter Diamessis

Sumedh M. Joshi, Peter J. Diamessis, Derek T. Steinmoeller et al.

A method for post-processing the velocity after a pressure projection is developed that helps to maintain stability in an under-resolved, inviscid, discontinuous element-based simulation for use in environmental fluid mechanics process studies. The post-processing method is needed because of spurious divergence growth at element interfaces due to the discontinuous nature of the discretization used. This spurious divergence eventually leads to a numerical instability. Previous work has shown that a discontinuous element-local projection onto the space of divergence-free basis functions is capable of stabilizing the projection method, but the discontinuity inherent in this technique may lead to instability in under-resolved simulations. By enforcing inter-element discontinuity and requiring a divergence-free result in the weak sense only, a new post-processing technique is developed that simultaneously improves smoothness and reduces divergence in the pressure-projected velocity field at the same time. When compared against a non-post-processed velocity field, the post-processed velocity field remains stable far longer and exhibits better smoothness and conservation properties.

Sumedh Joshi, Peter Diamessis

A combination of block-Jacobi and deflation preconditioning is used to solve a high-order discontinuous element-based collocation discretization of the Schur complement of the Poisson-Neumann system as arises in the operator splitting of the incompressible Navier-Stokes equations. The ill-posedness of the Poisson-Neumann system manifests as an inconsistency of the Schur complement problem, but it is shown that this can be accounted for with appropriate projections out of the null space of the Schur complement matrix without affecting the accuracy of the solution. The block-Jacobi preconditioner, combined with deflation, is shown to yield GMRES convergence independent of the polynomial order of expansion within an element. Finally, while the number of GMRES iterations does grow as the element size is reduced (e.g. $h$-refinement), the dependence is very mild; the number of GMRES iterations roughly doubles as the element size is divided by a factor of six. In light of these numerical results, the deflated Schur complement approach seems practicable, especially for high-order methods given its convergence independent of polynomial order.