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Multiscale and multilevel methods for porous media flow problems
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.
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

We consider two problems encountered in simulation of fluid flow through porous media. In macroscopic models based on Darcy's law, the permeability field appears as data.

The first problem is that the permeability field generally is not entirely known. We consider forward propagation of uncertainty from the permeability field to a quantity of interest. We focus on computing p-quantiles and failure probabilities of the quantity of interest. We propose and analyze improved standard and multilevel Monte Carlo methods that use computable error bounds for the quantity of interest. We show that substantial reductions in computational costs are possible by the proposed approaches.

The second problem is fine scale variations of the permeability field. The permeability often varies on a scale much smaller than that of the computational domain. For standard discretization methods, these fine scale variations need to be resolved by the mesh for the methods to yield accurate solutions. We analyze and prove convergence of a multiscale method based on the Raviart–Thomas finite element. In this approach, a low-dimensional multiscale space based on a coarse mesh is constructed from a set of independent fine scale patch problems. The low-dimensional space can be used to yield accurate solutions without resolving the fine scale.

Place, publisher, year, edition, pages
Uppsala University, 2015.
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2015-003
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-262276OAI: oai:DiVA.org:uu-262276DiVA: diva2:853228
Supervisors
Available from: 2015-09-09 Created: 2015-09-11 Last updated: 2017-08-31Bibliographically approved
List of papers
1. Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs
Open this publication in new window or tab >>Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs
2014 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 2, 826-850 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-242908 (URN)10.1137/140967039 (DOI)
Available from: 2014-12-23 Created: 2015-02-02 Last updated: 2017-12-05Bibliographically approved
2. A multilevel Monte Carlo method for computing failure probabilities
Open this publication in new window or tab >>A multilevel Monte Carlo method for computing failure probabilities
2016 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 4, 312-330 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262259 (URN)10.1137/140984294 (DOI)
Available from: 2016-04-05 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved
3. Improved Monte Carlo methods for computing failure probabilities of porous media flow systems
Open this publication in new window or tab >>Improved Monte Carlo methods for computing failure probabilities of porous media flow systems
2015 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2015-025
National Category
Computational Mathematics Oceanography, Hydrology, Water Resources
Identifiers
urn:nbn:se:uu:diva-261005 (URN)
Available from: 2015-08-31 Created: 2015-08-27 Last updated: 2016-07-05Bibliographically approved
4. Multiscale mixed finite elements
Open this publication in new window or tab >>Multiscale mixed finite elements
2016 (English)In: Discrete and Continuous Dynamical Systems. Series S, ISSN 1937-1632, E-ISSN 1937-1179, Vol. 9, 1269-1298 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262258 (URN)10.3934/dcdss.2016051 (DOI)000387662300002 ()
Available from: 2016-10-15 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved

Open Access in DiVA

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