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On Numerical Solution Methods for Block-Structured Discrete Systems
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.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The development, analysis, and implementation of efficient methods to solve algebraic systems of equations are main research directions in the field of numerical simulation and are the focus of this thesis. Due to their lesser demands for computer resources, iterative solution methods are the choice to make, when very large scale simulations have to be performed. To improve their efficiency, iterative methods are combined with proper techniques to accelerate convergence. A general technique to do this is to use a so-called preconditioner. Constructing and analysing various preconditioning methods has been an active field of research already for decades. Special attention is devoted to the class of the so-called optimal order preconditioners, that possess both optimal convergence rate and optimal computational complexity. The preconditioning techniques, proposed and studied in this thesis, utilise the block structure of the underlying matrices, and lead to methods that are of optimal order.

In the first part of the thesis, we construct an Algebraic MultiLevel Iteration (AMLI) method for systems arising from discretizations of parabolic problems, using Crouzeix-Raviart finite elements. The developed AMLI method is based on an approximated block factorization of the original system matrix, where the partitioning is associated with a sequence of nested discretization meshes.

In the second part of the thesis we develop solution methods for the numerical simulation of multiphase flow problems, modelled by the Cahn-Hilliard (C-H) equation. We consider the discrete C-H problem, obtained via finite element discretization in space and implicit schemes in time. We propose techniques to precondition the Jacobian of the discrete nonlinear system, based on its natural two-by-two block structure. The preconditioners are used in the framework of inexact Newton methods. We develop two nonlinear solution algorithms for the Cahn-Hilliard problem. Both lead to efficient optimal order methods. One of the main advantages of the proposed methods is that they are implemented using available software toolboxes for both sequential and distributed execution.

The theoretical analysis of the solution methods presented in this thesis is combined with numerical studies that confirm their efficiency.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. , 49 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 942
Keyword [en]
Preconditioning techniques, Finite element method, Two-by-two block matrices, Optimal order methods, AMLI method, Cahn-Hilliard equation, Multiphase flow, Inexact Newton method
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-173530ISBN: 978-91-554-8386-9 (print)OAI: oai:DiVA.org:uu-173530DiVA: diva2:523795
Public defence
2012-06-15, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2012-05-24 Created: 2012-04-26 Last updated: 2012-08-01Bibliographically approved
List of papers
1. Robust AMLI methods for parabolic Crouzeix–Raviart FEM systems
Open this publication in new window or tab >>Robust AMLI methods for parabolic Crouzeix–Raviart FEM systems
2010 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 235, 380-390 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-126208 (URN)10.1016/j.cam.2010.05.039 (DOI)000282708900005 ()
Available from: 2010-06-04 Created: 2010-06-07 Last updated: 2017-12-12Bibliographically approved
2. Block-preconditioners for conforming and non-conforming FEM discretizations of the Cahn–Hilliard equation
Open this publication in new window or tab >>Block-preconditioners for conforming and non-conforming FEM discretizations of the Cahn–Hilliard equation
2012 (English)In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2012, 549-557 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2012
Series
Lecture Notes in Computer Science, 7116
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-173521 (URN)10.1007/978-3-642-29843-1_62 (DOI)978-3-642-29842-4 (ISBN)
Available from: 2012-05-24 Created: 2012-04-25 Last updated: 2012-05-31Bibliographically approved
3. Efficient preconditioners for large scale binary Cahn–Hilliard models
Open this publication in new window or tab >>Efficient preconditioners for large scale binary Cahn–Hilliard models
2012 (English)In: Computational Methods in Applied Mathematics, ISSN 1609-4840, E-ISSN 1609-9389, Vol. 12, 1-22 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-173519 (URN)10.2478/cmam-2012-0001 (DOI)
Available from: 2012-03-02 Created: 2012-04-25 Last updated: 2017-12-07Bibliographically approved
4. Numerical and computational efficiency of solvers for two-phase problems
Open this publication in new window or tab >>Numerical and computational efficiency of solvers for two-phase problems
Show others...
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-002
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-167349 (URN)
Available from: 2012-01-23 Created: 2012-01-25 Last updated: 2012-05-24Bibliographically approved
5. Efficient numerical solution of discrete multi-component Cahn–Hilliard systems
Open this publication in new window or tab >>Efficient numerical solution of discrete multi-component Cahn–Hilliard systems
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-009
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-173529 (URN)
Available from: 2012-04-26 Created: 2012-04-26 Last updated: 2012-05-24Bibliographically approved

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