|dc.description.abstract||The incompressible Navier-Stokes equations are of major interest due to their importance in modelling fluid flow problems. However, solving the Navier-Stokes equations is a difficult task. To address this problem, in this thesis, we consider fast and efficient solvers. We are particularly interested in solving a new class of hybridizable discontinuous Galerkin (HDG) discretizations of the incompressible Navier-Stokes equations, as these discretizations result in exact mass conservation, are locally conservative, and have fewer degrees of freedom than discontinuous Galerkin methods (which is typically used for advection dominated flows). To achieve this goal, we have made various contributions to related problems, as I discuss next.
Firstly, we consider the solution of matrices with 2x2 block structure. We are interested in this problem as many discretizations of the Navier-Stokes equations result in block linear systems of equations, especially discretizations based on mixed-finite element methods like HDG. These systems also arise in other areas of computational mathematics, such as constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, we prove a direct equivalence between convergence of 2x2 block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of 2x2 block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. We verify the theoretical results numerically on an HDG discretization of the steady linearized Navier--Stokes equations. The findings also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses.
Secondly, as an initial step towards solving the time-dependent Navier-Stokes equations, we investigate the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time HDG discretization of the scalar advection-diffusion equation. The motivation for this study is two-fold. First, the HDG discretization of the velocity part of the momentum block of the linearized Navier-Stokes equations is the HDG discretization of the vector advection-diffusion equation. Hence, efficient and fast solution of the advection-diffusion problem is a prerequisite for developing fast solvers for the Navier-Stokes equations. The second reason to study this all-at-once space-time problem is that the time-dependent advection-diffusion equation can be seen as a ``steady'' advection-diffusion problem in (d+1)-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. We present numerical examples which demonstrate the effectiveness of AIR as a preconditioner for time-dependent advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems.
As the final topic of this thesis, we extend two state-of-the-art preconditioners for the Navier-Stokes equations, namely, the pressure convection-diffusion and the grad-div/augmented Lagrangian preconditioners to HDG discretizations. Our preconditioners are simple to implement, and our numerical results show that these preconditioners are robust in h and only mildly dependent on the Reynolds numbers.||en