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Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
2010 (English)In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2010, 108-115 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag , 2010. 108-115 p.
Series
Lecture Notes in Computer Science, 5910
National Category
Computational Mathematics Computer Science
Identifiers
URN: urn:nbn:se:uu:diva-125577DOI: 10.1007/978-3-642-12535-5_11ISI: 000278091900011ISBN: 978-3-642-12534-8 (print)OAI: oai:DiVA.org:uu-125577DiVA: diva2:320240
Available from: 2010-05-10 Created: 2010-05-24 Last updated: 2012-09-12Bibliographically approved
In thesis
1. Robust preconditioning methods for algebraic problems, arising in multi-phase flow models
Open this publication in new window or tab >>Robust preconditioning methods for algebraic problems, arising in multi-phase flow models
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The aim of the project is to construct, analyse and implement fast and reliable numerical solution methods to simulate multi-phase flow, modeled by a coupled system consisting of the time-dependent Cahn-Hilliard and incompressible Navier-Stokes equations with variable viscosity and variable density. This thesis mainly discusses the efficient solution methods for the latter equations aiming at constructing preconditioners, which are numerically and computationally efficient, and robust with respect to various problem, discretization and method parameters.

In this work we start by considering the stationary Navier-Stokes problem with constant viscosity. The system matrix arising from the finite element discretization of the linearized Navier-Stokes problem is nonsymmetric of saddle point form, and solving systems with it is the inner kernel of the simulations of numerous physical processes, modeled by the Navier-Stokes equations. Aiming at reducing the simulation time, in this thesis we consider iterative solution methods with efficient preconditioners. When discretized with the finite element method, both the Cahn-Hilliard equations and the stationary Navier-Stokes equations with constant viscosity give raise to linear algebraic systems with nonsymmetric matrices of two-by-two block form. In Paper I we study both problems and apply a common general framework to construct a preconditioner, based on the matrix structure. As a part of the general framework, we use the so-called element-by-element Schur complement approximation. The implementation of this approximation is rather cheap. However, the numerical experiments, provided in the paper, show that the preconditioner is not fully robust with respect to the problem and discretization parameters, in this case the viscosity and the mesh size. On the other hand, for not very convection-dominated flows, i.e., when the viscosity is not very small, this approximation does not depend on the mesh size and works efficiently. Considering the stationary Navier-Stokes equations with constant viscosity, aiming at finding a preconditioner which is fully robust to the problem and discretization parameters, in Paper II we turn to the so-called augmented Lagrangian (AL) approach, where the linear system is transformed into an equivalent one and then the transformed system is iteratively solved with the AL type preconditioner. The analysis in Paper II focuses on two issues, (1) the influence of a scalar method parameter (a stabilization constant in the AL method) on the convergence rate of the preconditioned method and (2) the choice of a matrix parameter for the AL method, which involves an approximation of the inverse of the finite element mass matrix. In Paper III we consider the stationary Navier-Stokes problem with variable viscosity. We show that the known efficient preconditioning techniques in particular, those for the AL method, derived for constant viscosity, can be straightforwardly applicable also in this case.

One often used technique to solve the incompressible Navier-Stokes problem with variable density is via operator splitting, i.e., decoupling of the solutions for density, velocity and pressure. The operator splitting technique introduces an additional error, namely the splitting error, which should be also considered, together with discretization errors in space and time. Insuring the accuracy of the splitting scheme usually induces additional constrains on the size of the time-step. Aiming at fast numerical simulations and using large time-steps may require to use higher order time-discretization methods. The latter issue and its impact on the preconditioned iterative solution methods for the arising linear systems are envisioned as possible directions for future research.

When modeling multi-phase flows, the Navier-Stokes equations should be considered in their full complexity, namely, the time-dependence, variable viscosity and variable density formulation. Up to the knowledge of the author, there are not many studies considering all aspects simultaneously. Issues on this topic, in particular on the construction of efficient preconditioners of the arising matrices need to be further studied.

Place, publisher, year, edition, pages
Uppsala University, 2011
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2011-002
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-151683 (URN)
Supervisors
Available from: 2011-04-18 Created: 2011-04-15 Last updated: 2017-08-31Bibliographically approved
2. On some Numerical Methods and Solution Techniques for Incompressible Flow Problems
Open this publication in new window or tab >>On some Numerical Methods and Solution Techniques for Incompressible Flow Problems
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. 56 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 954
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-179410 (URN)978-91-554-8429-3 (ISBN)
Public defence
2012-09-24, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 14:00 (English)
Opponent
Supervisors
Available from: 2012-09-03 Created: 2012-08-14 Last updated: 2013-01-22Bibliographically approved

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