Preconditioning of Hybridizable Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

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

Üç boyutlu kısmi diferansiyel denklemlerin numerik çözümü için paralel önkoşullandırma teknikleri.

Partial differential equations are commonly used in industry and science to model observed phenomena and gain insight regarding phenomena or solve related problems. Recently three dimensional partial differential equations started to become more and more essential and popular. Numerical solution of these problems usually is composed of two steps; discretization with some scheme and solving resulting sparse linear system which is large and usually ill-conditioned. Large size encourages usage of iterative solvers rather that direct solvers due to small memory requirement and short solution times, but iterative solvers mostly fail for ill-conditioned coefficient matrices which is a great discouragement. Preconditioning is a remedy for this problem. Solution of large sparse linear systems take large amounts of time and usually consecutive solution of linear systems is necessary. Parallel computing techniques are used to overcome this problem. Various preconditioning techniques with iterative solutions and their scalability on three different parallel computing platforms are investigated for the solution of two three dimensional partial differential equation related large scale problems which are important for industrial applications and scientific modelling. Results of this investigation are compared against direct solvers and each other.M.S. - Master of Scienc

Improved Parallel Preconditioners for Multiphysics Topology Optimizations

Topology optimization, also known as layout optimization, involves multiple physics encompassing structural, thermal, fluidic electrical and electromechanical systems including coupled phenomena such as solids and fluids as in convective cooling systems, aerodynamical systems, electronics, actuators and motors. In the root of topology optimization is the repeated solution of the finite element equations Au = f representing the physics of the problem at hand such as elasticity, heat transfer, fluid flow and electromagnetics, where A is the coefficient matrix, which is highly sparse, u is the vector of physical unknowns (displacements, temperature or velocity, etc.), f is the known source (load) vector. Numerical difficulties associated with these equations are two folds. Firstly, they are very large systems, due to large number of cells (finite elements) needed for defining a topology, requiring the use of the iterative solvers such as the conjugate gradient algorithm instead of the direct solvers. Even with that, the solution times tend to become very high, which is often tried to be resolved by parallel solution strategies. Secondly, they are extremely ill-conditioned due to highly heterogeneous material distributions that evolve during the course of topology formations, with material properties of elements varying from nearly zero values in empty regions to very large values in full regions, which is usually addressed by matrix preconditioning. When simple preconditioners, such as diagonal and incomplete LU factorization preconditioners, are used parallel efficiency deteriorate very fast due to increased number of iterations needed for convergence as more processors are used. In this paper, we modify our previous parallel topology solver implementation by introducing new matrix reordering and scaling schemes to improve the scalability of iterative solvers. We use here more sophisticated preconditioners in order to improve the parallel scalability of the iterative algorithm, even for ill-conditioned cases; not only due to heterogeneous material properties, but also due to condition number deterioration in the equations of coupled multiphysics media. The developed solver was tested for accuracy and parallel efficiency of extreme cases, demonstrating high parallel scalability

Discretization and parallel iterative schemes for advection-diffusion-reaction problems

\u3cp\u3eConservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For discretization we use the finite volume method in combination with the complete flux scheme. The numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and the inhomogeneous flux, taking into account the effect of the source term (ten Thije Boonkkamp and Anthonissen, J Sci Comput 46(1):47–70, 2011). For a three-dimensional conservation law this results in a 27-point coupling for the unknown as well as the source term. Direct solution of the sparse linear systems that arise in 3D ADR problems is not feasible due to fill-in. Iterative solution of such linear systems remains to be the only efficient alternative which requires less memory and shorter time to solution compared to direct solvers. Iterative solvers require a preconditioner to reduce the number of iterations. We present results using several different preconditioning techniques and study their effectiveness.\u3c/p\u3

Discretization and Parallel Iterative Schemes for Advection-Diffusion-Reaction Problems

Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For discretization we use the finite volume method in combination with the complete flux scheme. The numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and the inhomogeneous flux, taking into account the effect of the source term (ten Thije Boonkkamp and Anthonissen, J Sci Comput 46(1): 47-70, 2011). For a three-dimensional conservation law this results in a 27point coupling for the unknown as well as the source term. Direct solution of the sparse linear systems that arise in 3D ADR problems is not feasible due to fill-in. Iterative solution of such linear systems remains to be the only efficient alternative which requires less memory and shorter time to solution compared to direct solvers. Iterative solvers require a preconditioner to reduce the number of iterations. We present results using several different preconditioning techniques and study their effectiveness