uu.seUppsala University Publications

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On Numerical Solution Methods for Block-Structured Discrete SystemsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis, 2012. , p. 49
##### Series

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 942
##### Keywords [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, id: diva2:523795
##### Public defence

2012-06-15, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt831",{id:"formSmash:j_idt831",widgetVar:"widget_formSmash_j_idt831",multiple:true}); Available from: 2012-05-24 Created: 2012-04-26 Last updated: 2012-08-01Bibliographically approved
##### List of papers

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.

1. Robust AMLI methods for parabolic Crouzeix–Raviart FEM systems$(function(){PrimeFaces.cw("OverlayPanel","overlay322449",{id:"formSmash:j_idt925:0:j_idt935",widgetVar:"overlay322449",target:"formSmash:j_idt925:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Block-preconditioners for conforming and non-conforming FEM discretizations of the Cahn–Hilliard equation$(function(){PrimeFaces.cw("OverlayPanel","overlay523764",{id:"formSmash:j_idt925:1:j_idt935",widgetVar:"overlay523764",target:"formSmash:j_idt925:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Efficient preconditioners for large scale binary Cahn–Hilliard models$(function(){PrimeFaces.cw("OverlayPanel","overlay523699",{id:"formSmash:j_idt925:2:j_idt935",widgetVar:"overlay523699",target:"formSmash:j_idt925:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Numerical and computational efficiency of solvers for two-phase problems$(function(){PrimeFaces.cw("OverlayPanel","overlay483790",{id:"formSmash:j_idt925:3:j_idt935",widgetVar:"overlay483790",target:"formSmash:j_idt925:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Efficient numerical solution of discrete multi-component Cahn–Hilliard systems$(function(){PrimeFaces.cw("OverlayPanel","overlay523767",{id:"formSmash:j_idt925:4:j_idt935",widgetVar:"overlay523767",target:"formSmash:j_idt925:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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