uu.seUppsala University Publications

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Robust Preconditioners Based on the Finite Element FrameworkPrimeFaces.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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2007 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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

Uppsala: Acta Universitatis Upsaliensis , 2007. , p. 84
##### Series

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 296
##### Keywords [en]

FEM, iterative solution method, algebraic multilevel preconditioner, sparse approximate inverse, block preconditioner, Schur complement approximation, nonsymmetric saddle point matrix, isostatic glacial adjustment, pre-stress advection, elasticity, viscoelasticity, (in)compressible solid, ABAQUS, BEM/DDM
##### National Category

Computational Mathematics
##### Research subject

Scientific Computing
##### Identifiers

URN: urn:nbn:se:uu:diva-7828ISBN: 978-91-554-6870-5 (print)OAI: oai:DiVA.org:uu-7828DiVA, id: diva2:170126
##### Public defence

2007-05-11, Room 2247, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true}); Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2011-10-26Bibliographically approved
##### List of papers

Robust preconditioners on block-triangular and block-factorized form for three types of linear systems of two-by-two block form are studied in this thesis.

The first type of linear systems, which are dense, arise from a boundary element type of discretization of crack propagation problems. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method.

The second type of algebraic systems, which are sparse, indefinite and nonsymmetric, arise from a finite element (FE) discretization of the partial differential equations (PDE) that describe (visco)elastic glacial isostatic adjustment (GIA). The Schur complement approximation in the block preconditioners is constructed by assembly of local, exactly computed Schur matrices. The quality of the approximation is verified in numerical experiments.

When the block preconditioners for the indefinite problem are combined with an inner iterative scheme preconditioned by a (nearly) optimal multilevel preconditioner, the resulting preconditioner is (nearly) optimal and robust with respect to problem size, material parameters, number of space dimensions, and coefficient jumps.

Two approaches to mathematically formulate the PDEs for GIA are compared. In the first approach the equations are formulated in their full complexity, whereas in the second their formulation is confined to the features and restrictions of the employed FE package. Different solution methods for the algebraic problem are used in the two approaches. Analysis and numerical experiments reveal that the first strategy is more accurate and efficient than the latter.

The block structure in the third type of algebraic systems is due to a fine-coarse splitting of the unknowns. The inverse of the pivot block is approximated by a sparse matrix which is assembled from local, exactly inverted matrices. Numerical experiments and analysis of the approximation show that it is robust with respect to problem size and coefficient jumps.

1. Algebraic preconditioning versus direct solvers for dense linear systems as arising in crack propagation problems$(function(){PrimeFaces.cw("OverlayPanel","overlay101700",{id:"formSmash:j_idt495:0:j_idt499",widgetVar:"overlay101700",target:"formSmash:j_idt495:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Numerical simulations of glacial rebound using preconditioned iterative solution methods$(function(){PrimeFaces.cw("OverlayPanel","overlay101697",{id:"formSmash:j_idt495:1:j_idt499",widgetVar:"overlay101697",target:"formSmash:j_idt495:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. An agglomerate multilevel preconditioner for linear isostasy saddle point problems$(function(){PrimeFaces.cw("OverlayPanel","overlay106900",{id:"formSmash:j_idt495:2:j_idt499",widgetVar:"overlay106900",target:"formSmash:j_idt495:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Preconditioning of nonsymmetric saddle point systems as arising in modelling of viscoelastic problems$(function(){PrimeFaces.cw("OverlayPanel","overlay40154",{id:"formSmash:j_idt495:3:j_idt499",widgetVar:"overlay40154",target:"formSmash:j_idt495:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. A comparison between two solution techniques to solve the equations of glacially induced deformation of an elastic Earth$(function(){PrimeFaces.cw("OverlayPanel","overlay41373",{id:"formSmash:j_idt495:4:j_idt499",widgetVar:"overlay41373",target:"formSmash:j_idt495:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Finite element block-factorized preconditioners$(function(){PrimeFaces.cw("OverlayPanel","overlay38399",{id:"formSmash:j_idt495:5:j_idt499",widgetVar:"overlay38399",target:"formSmash:j_idt495:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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