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Analysis and Implementation of Preconditioners for Prestressed Elasticity Problems: Advances and Enhancements
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.ORCID iD: 0000-0003-0985-0741
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

In this work, prestressed elasticity problem as a model of the so-called glacial isostatic adjustment (GIA) process is studied. The model problem is described by a set of partial differential equations (PDE) and discretized with a mixed finite element (FE) formulation. In the presence of prestress the so-constructed system of equations is non-symmetric and indefinite. Moreover, the resulting system of equations is of the saddle point form.

We focus on a robust and efficient block lower-triangular preconditioning method, where the lower diagonal block is and approximation of the so-called Schur complement. The Schur complement is approximated by the so-called element-wise Schur complement. The element-wise Schur complement is constructed by assembling exact local Schur complements on the cell elements and distributing the resulting local matrices to the global preconditioner matrix.

We analyse the properties of the element-wise Schur complement for the symmetric indefinite system matrix and provide proof of its quality. We show that the spectral radius of the element-wise Schur complement is bounded by the exact Schur complement and that the quality of the approximation is not affected by the domain shape.

The diagonal blocks of the lower-triangular preconditioner are combined with inner iterative schemes accelerated by (numerically) optimal and robust algebraic multigrid (AMG) preconditioner. We observe that on distributed memory systems, the top pivot block of the preconditioner is not scaling satisfactorily. The implementation of the methods is further studied using a general profiling tool, designed for clusters.

For nonsymmetric matrices we use the theory of Generalized Locally Toeplitz (GLT) matrices and show the spectral behavior of the element-wise Schur complement, compared to the exact Schur complement. Moreover, we use the properties of the GLT matrices to construct a more efficient AMG preconditioner. Numerical experiments show that the so-constructed methods are robust and optimal.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2017. , 61 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1580
Keyword [en]
FEM, Saddle point matrix, Preconditioning, Schur complement, Generalized Locally Toeplitz, Prestressed elasticity
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
URN: urn:nbn:se:uu:diva-331852ISBN: 978-91-513-0116-7 (print)OAI: oai:DiVA.org:uu-331852DiVA: diva2:1150391
Public defence
2017-12-08, Room 2446, TDB, Lägerhyddsvägen 2, 75237, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-11-13 Created: 2017-10-18 Last updated: 2017-11-13
List of papers
1. Numerical and computational aspects of some block-preconditioners for saddle point systems
Open this publication in new window or tab >>Numerical and computational aspects of some block-preconditioners for saddle point systems
2015 (English)In: Parallel Computing, ISSN 0167-8191, E-ISSN 1872-7336, Vol. 49, 164-178 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-253717 (URN)10.1016/j.parco.2015.06.003 (DOI)000364892500012 ()
Available from: 2015-06-26 Created: 2015-06-01 Last updated: 2017-10-18Bibliographically approved
2. Spectral analysis of coupled PDEs and of their Schur complements via Generalized Locally Toeplitz sequences in 2D
Open this publication in new window or tab >>Spectral analysis of coupled PDEs and of their Schur complements via Generalized Locally Toeplitz sequences in 2D
2016 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 309, 74-105 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-296265 (URN)10.1016/j.cma.2016.05.042 (DOI)000383828400004 ()
Available from: 2016-06-08 Created: 2016-06-14 Last updated: 2017-10-18Bibliographically approved
3. Schur complement matrix and its (elementwise) approximation: A spectral analysis based on GLT sequences
Open this publication in new window or tab >>Schur complement matrix and its (elementwise) approximation: A spectral analysis based on GLT sequences
2015 (English)In: Large-Scale Scientific Computing, Springer, 2015, 419-426 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Springer, 2015
Series
Lecture Notes in Computer Science, 9374
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-267963 (URN)10.1007/978-3-319-26520-9_47 (DOI)000373881700047 ()978-3-319-26519-3 (ISBN)
Conference
LSSC 2015
Funder
Knut and Alice Wallenberg Foundation
Available from: 2015-11-29 Created: 2015-11-30 Last updated: 2017-10-18Bibliographically approved
4. Multidimensional performance and scalability analysis for diverse applications based on system monitoring data
Open this publication in new window or tab >>Multidimensional performance and scalability analysis for diverse applications based on system monitoring data
Show others...
2017 (English)In: Parallel Processing and Applied Mathematics, Springer, 2017Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Springer, 2017
Series
Lecture Notes in Computer Science
National Category
Computer Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-331839 (URN)
Conference
PPAM 2017
Projects
eSSENCE
Note

to appear

Available from: 2017-10-18 Created: 2017-10-18 Last updated: 2017-10-20Bibliographically approved
5. Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem
Open this publication in new window or tab >>Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem
Show others...
2017 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, 1015-1028 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-323893 (URN)10.1016/j.camwa.2017.05.024 (DOI)
Funder
Knut and Alice Wallenberg Foundation
Available from: 2017-06-03 Created: 2017-06-10 Last updated: 2017-10-18Bibliographically approved
6. Function-based algebraic multigrid method for the 3D Poisson problem on structured meshes
Open this publication in new window or tab >>Function-based algebraic multigrid method for the 3D Poisson problem on structured meshes
2017 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2017-022
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-331778 (URN)
Available from: 2017-10-18 Created: 2017-10-18 Last updated: 2017-10-19Bibliographically approved

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