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Robust preconditioning methods for algebraic problems, arising in multi-phase flow models
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.
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 universitet, 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: urn:nbn:se:uu:diva-151683OAI: oai:DiVA.org:uu-151683DiVA: diva2:410931
Supervisors
Available from: 2011-04-18 Created: 2011-04-15 Last updated: 2014-08-06Bibliographically approved
List of papers
1. Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
Open this publication in new window or tab >>Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form
2010 (English)In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2010, 108-115 p.Conference paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2010
Series
, Lecture Notes in Computer Science, 5910
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-125577 (URN)10.1007/978-3-642-12535-5_11 (DOI)000278091900011 ()978-3-642-12534-8 (ISBN)
Available from: 2010-05-10 Created: 2010-05-24 Last updated: 2012-09-12Bibliographically approved
2. On an augmented Lagrangian-based preconditioning of Oseen type problems
Open this publication in new window or tab >>On an augmented Lagrangian-based preconditioning of Oseen type problems
2011 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, 865-888 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-156201 (URN)10.1007/s10543-011-0334-4 (DOI)000297362000005 ()
Available from: 2011-06-07 Created: 2011-07-17 Last updated: 2016-01-28Bibliographically approved
3. Preconditioning the incompressible Navier-Stokes equations with variable viscosity
Open this publication in new window or tab >>Preconditioning the incompressible Navier-Stokes equations with variable viscosity
2011 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-006
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-151675 (URN)
Available from: 2011-04-05 Created: 2011-04-15 Last updated: 2011-11-18Bibliographically approved

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