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  • 1. Axelsson, Owe
    et al.
    He, Xin
    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.
    Neytcheva, Maya
    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.
    Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity2012Report (Other academic)
  • 2.
    Axelsson, Owe
    et al.
    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.
    He, Xin
    Neytcheva, Maya
    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.
    Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity: Part I2015In: Mathematical Modelling and Analysis, ISSN 1392-6292, E-ISSN 1648-3510, Vol. 20, p. 232-260Article in journal (Refereed)
  • 3.
    He, Xin
    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.
    On some Numerical Methods and Solution Techniques for Incompressible Flow Problems2012Doctoral 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.

    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, p. 108-115Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2010
    Series
    Lecture Notes in Computer Science ; 5910
    National Category
    Computational Mathematics Computer Sciences
    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: 2018-01-12Bibliographically 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, p. 865-888Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    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: 2019-01-22Bibliographically 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
    2012 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 30, p. 461-482Article in journal (Refereed) Published
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-179408 (URN)10.4208/jcm.1201-m3848 (DOI)000311467400002 ()
    Available from: 2012-09-24 Created: 2012-08-14 Last updated: 2017-12-07Bibliographically approved
    4. Efficiently parallel implementation of the inverse Sherman–Morrison algorithm
    Open this publication in new window or tab >>Efficiently parallel implementation of the inverse Sherman–Morrison algorithm
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-017
    National Category
    Computer Sciences Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-179219 (URN)
    Projects
    UPMARCeSSENCE
    Available from: 2012-08-08 Created: 2012-08-09 Last updated: 2018-01-12Bibliographically approved
    5. On preconditioning incompressible non-Newtonian flow problems
    Open this publication in new window or tab >>On preconditioning incompressible non-Newtonian flow problems
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-016
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-179218 (URN)
    Available from: 2012-08-07 Created: 2012-08-09 Last updated: 2018-01-12Bibliographically approved
    6. Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity
    Open this publication in new window or tab >>Numerical solution of the time-dependent Navier–Stokes equation for variable density–variable viscosity
    2012 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-019
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-179351 (URN)
    Available from: 2012-08-12 Created: 2012-08-13 Last updated: 2012-09-12Bibliographically approved
  • 4.
    He, Xin
    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.
    Robust preconditioning methods for algebraic problems, arising in multi-phase flow models2011Licentiate 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.

    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, p. 108-115Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2010
    Series
    Lecture Notes in Computer Science ; 5910
    National Category
    Computational Mathematics Computer Sciences
    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: 2018-01-12Bibliographically 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, p. 865-888Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    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: 2019-01-22Bibliographically 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 Sciences
    Identifiers
    urn:nbn:se:uu:diva-151675 (URN)
    Available from: 2011-04-05 Created: 2011-04-15 Last updated: 2018-01-12Bibliographically approved
  • 5.
    He, Xin
    et al.
    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.
    Holm, Marcus
    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, Computational Science.
    Neytcheva, Maya
    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.
    Efficiently parallel implementation of the inverse Sherman–Morrison algorithm2012Report (Other academic)
  • 6.
    He, Xin
    et al.
    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.
    Holm, Marcus
    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, Computational Science.
    Neytcheva, Maya
    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.
    Parallel implementation of the Sherman–Morrison matrix inverse algorithm2013In: Applied Parallel and Scientific Computing, Berlin: Springer-Verlag , 2013, p. 206-219Conference paper (Refereed)
  • 7.
    He, Xin
    et al.
    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.
    Neytcheva, Maya
    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.
    On preconditioning incompressible non-Newtonian flow problems2012Report (Other academic)
  • 8.
    He, Xin
    et al.
    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.
    Neytcheva, Maya
    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.
    Preconditioning the incompressible Navier-Stokes equations with variable viscosity2012In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 30, p. 461-482Article in journal (Refereed)
  • 9.
    He, Xin
    et al.
    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.
    Neytcheva, Maya
    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.
    Preconditioning the incompressible Navier-Stokes equations with variable viscosity2011Report (Other academic)
  • 10.
    He, Xin
    et al.
    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.
    Neytcheva, Maya
    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.
    Serra Capizzano, Stefano
    On an augmented Lagrangian-based preconditioning of Oseen type problems2011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, p. 865-888Article in journal (Refereed)
  • 11.
    He, Xin
    et al.
    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.
    Neytcheva, Maya
    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.
    Serra Capizzano, Stefano
    On an augmented Lagrangian-based preconditioning of Oseen type problems2010Report (Other academic)
  • 12. He, Xin
    et al.
    Neytcheva, Maya
    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.
    Vuik, Cornelis
    On preconditioning of incompressible non-Newtonian flow problems2015In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 33, p. 33-58Article in journal (Refereed)
  • 13.
    Neytcheva, Maya
    et al.
    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.
    Do-Quang, Minh
    He, Xin
    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.
    Element-by-element Schur complement approximations for general nonsymmetric matrices of two-by-two block form2010In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2010, p. 108-115Conference paper (Refereed)
1 - 13 of 13
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