On some Numerical Methods and Solution Techniques for Incompressible Flow Problems

University dissertation from Uppsala : Acta Universitatis Upsaliensis

Abstract: The focus of this work is on numerical solution methods for solving the incompressible Navier-Stokes equations, which consist of a set of coupled nonlinear partial differential equations.In general, after linearization and finite element discretization in space, the original nonlinear problem is converted into finding the solutions of a sequence of linear systems of equations. Because of the underlying mathematical model, the coefficient matrix of the linear system is indefinite and nonsymmetric of two-by-two block structure. Due to their less demands for computer resources than direct methods, iterative solution methods are chosen to solve these linear systems. In order to accelerate the convergence rate of the iterative methods, efficient preconditioning techniques become essential. How to construct numerically efficient preconditioners for two-by-two block systems arising in the incompressible Navier-Stokes equations has been studied intensively during the past decades, and is also a main concern in this thesis.The Navier-Stokes equations depend on various problem parameters, such as density and viscosity, that themselves may vary in time and space as in multiphase systems. In this thesis we follow the following strategy. First, we consider the stationary Navier-Stokes equations with constant viscosity and density, and contribute to the search of efficient preconditioners by analyzing and testing the element-by-element approximation method of the Schur complement matrix and the so-called augmented Lagrangian method. Second, the variation of the viscosity is an important factor and affects the behavior of the already known preconditioners, proposed for two-by-two block matrices. To this end, we choose the augmented Lagrangian method and analyse the impact of the variation of the viscosity on the resulting preconditioner. Finally, we consider the Navier-Stokes equations with their full complexity, namely, time dependence, variable density and variable viscosity. Fast and reliable solution methods are constructed based on a reformulation of the original equations and some operator splitting techniques. Preconditioners for the so-arising linear systemsare also analyzed and tested.