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

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.

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
Supervisors
Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2011-10-26Bibliographically approved
List of papers
1. Algebraic preconditioning versus direct solvers for dense linear systems as arising in crack propagation problems
Open this publication in new window or tab >>Algebraic preconditioning versus direct solvers for dense linear systems as arising in crack propagation problems
2005 (English)In: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 21, p. 73-81Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-73790 (URN)10.1002/cnm.728 (DOI)
Available from: 2006-05-13 Created: 2006-05-13 Last updated: 2018-01-14Bibliographically approved
2. Numerical simulations of glacial rebound using preconditioned iterative solution methods
Open this publication in new window or tab >>Numerical simulations of glacial rebound using preconditioned iterative solution methods
2005 (English)In: Applications of Mathematics, ISSN 0862-7940, E-ISSN 1572-9109, Vol. 50, p. 183-201Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-73787 (URN)10.1007/s10492-005-0013-3 (DOI)
Available from: 2006-05-13 Created: 2006-05-13 Last updated: 2018-01-14Bibliographically approved
3. An agglomerate multilevel preconditioner for linear isostasy saddle point problems
Open this publication in new window or tab >>An agglomerate multilevel preconditioner for linear isostasy saddle point problems
2006 (English)In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2006, p. 113-120Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2006
Series
Lecture Notes in Computer Science ; 3743
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-78987 (URN)10.1007/11666806_11 (DOI)000236456400011 ()
Available from: 2008-07-22 Created: 2008-07-22 Last updated: 2018-01-13Bibliographically approved
4. Preconditioning of nonsymmetric saddle point systems as arising in modelling of viscoelastic problems
Open this publication in new window or tab >>Preconditioning of nonsymmetric saddle point systems as arising in modelling of viscoelastic problems
2008 (English)In: Electronic Transactions on Numerical Analysis, ISSN 1068-9613, E-ISSN 1068-9613, Vol. 29, p. 193-211Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-12385 (URN)000207070800013 ()
Available from: 2007-12-17 Created: 2007-12-17 Last updated: 2018-01-12Bibliographically approved
5. A comparison between two solution techniques to solve the equations of glacially induced deformation of an elastic Earth
Open this publication in new window or tab >>A comparison between two solution techniques to solve the equations of glacially induced deformation of an elastic Earth
2008 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 75, p. 479-502Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-13603 (URN)10.1002/nme.2268 (DOI)000258113600004 ()
Available from: 2008-07-14 Created: 2008-07-14 Last updated: 2018-01-12Bibliographically approved
6. Finite element block-factorized preconditioners
Open this publication in new window or tab >>Finite element block-factorized preconditioners
2007 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2007-008
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-10631 (URN)
Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2011-11-18Bibliographically approved

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