Abdullah Ali Sivas | Applied Math, University of Waterloo
Preconditioning of Hybridized Discontinuous Galerkin Discretizations of the Incompressible Navier-Stokes Equations
The Navier-Stokes equations are useful in modeling many practical and real-life applications such as fluid flow around wings, wind turbines, ship hulls; and blood flow inside the body. Because of this, fast solution techniques for these equations are important.
For engineering applications, the numerical discretization of the Navier-Stokes equations typically results in large linear systems of equations, which are challenging to solve. To solve these linear systems, we will focus on iterative solution methods. For many problems, convergence of iterative solvers is slow. We address this problem by employing specialized preconditioners. The purpose of preconditioners is to change the coefficient matrix to one that is easier to solve using iterative methods. Usually, the performance of a preconditioner depends on the mesh size: as one refines the mesh to get a more accurate solution, the performance of the preconditioner degrades. Optimal preconditioners are a special kind of preconditioner which do not suffer from this phenomenon. We will show the derivation of a known optimal preconditioner for the Stokes equations and demonstrate its efficiency to motivate our research.
There are no known optimal preconditioners for discretizations of the Navier-Stokes equations, although non-optimal preconditioners which perform well for low Reynolds numbers exist for some discretizations. In this talk we will review the Pressure Convection-Diffusion (PCD)  preconditioner for standard Taylor-Hood finite element methods.
The main goal of this talk, however, will be the generalization of this preconditioner to Hybridized Discontinuous Galerkin (HDG) methods for the approximate solution of the incompressible Navier-Stokes Equations. HDG methods are a new class of finite element method, which have many appealing properties. The HDG method of , for example, is pressure-robust, i.e., the error in the velocity does not depend on the error in the pressure. This is especially important for high-Reynolds number flows. We will present preliminary results of our investigation. We will then outline several approaches to find an efficient, scalable and ideally an optimal preconditioner for HDG discretizations of the Navier-Stokes equations.
 Elman, H. C., Silvester, D. J., & Wathen, A. J. (2014). Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation.
 Rhebergen, S. & Wells, G.N. J Sci Comput (2018). https://doi.org/10.1007/s10915-018-0671-4