The numerical solution of the pressure Poisson equation for the incompressible Navier-Stokes equations using a quadrilateral spectral multidomain penalty method

We outline the basic features of a spectral multidomain penalty method (SMPM)based solver for the pressure Poisson equation (PPE) with Neumann boundary conditions, as encountered in the time-discretization of the incompressible Navier-Stokes equations. One one hand, the SMPM discretization enables robust under-resolved simulations without sacrificing high accuracy. On the other, the solution of the PPE is an inevitable requirement when simulating strongly non-hydrostatic flows, such as those occuring in the natural environment. The fundamental building blocks of the PPE solver presented here are a Kronecker (tensor) product-based computation of the left null singular value of the nonsymmetric SMPM-discretized Laplacian matrix and a custom-designed two-level preconditioner. Both of these tools are essential towards ensuring existence and uniqueness of the solution of the discrete linear system of equations and enabling its efficient iterative calculation. The accuracy and efficiency of the PPE solver are demonstrated through application to two incompressible flow benchmarks, the Taylor vortex and the Lid-driven cavity flow. In addition to providing an efficient tool for the iterative solution of the PPE in incompressible flow simulations, this work presents algorithms which are of interest to the subdiscipline of the numerical linear algebra community focused on the iterative solution of consistent singular non-symmetric linear systems of equations.