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An adaptive discontinuous Galerkin multiscale method for elliptic 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.
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.
2013 (English)In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 11, 747-765 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
2013. Vol. 11, 747-765 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-200254DOI: 10.1137/120863162ISI: 000325006000003OAI: oai:DiVA.org:uu-200254DiVA: diva2:622894
Available from: 2013-08-01 Created: 2013-05-23 Last updated: 2017-12-06Bibliographically approved
In thesis
1. On discontinuous Galerkin multiscale methods
Open this publication in new window or tab >>On discontinuous Galerkin multiscale methods
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis a new multiscale method, the discontinuous Galerkin multiscale method, is proposed. The method uses localized fine scale computations to correct a global coarse scale equation and thereby takes the fine scale features into account. We show a priori error bounds for convection dominated diffusion-convection-reaction problems with variable coefficients. We also present a posteriori error bound in the case of no convection or reaction and present an adaptive algorithm which tunes the method parameters automatically. We also present extensive numerical experiments which verify our analytical findings.

Place, publisher, year, edition, pages
Uppsala University, 2013
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2013-003
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-200260 (URN)
Supervisors
Available from: 2013-06-04 Created: 2013-05-23 Last updated: 2017-08-31Bibliographically approved
2. Multiscale Methods and Uncertainty Quantification
Open this publication in new window or tab >>Multiscale Methods and Uncertainty Quantification
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements.

We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries.

For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2015. 32 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1287
Keyword
multiscale methods, finite element method, discontinuous Galerkin, Petrov-Galerkin, a priori, a posteriori, complex geometry, uncertainty quantification, multilevel Monte Carlo, failure probability
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-262354 (URN)978-91-554-9336-3 (ISBN)
Public defence
2015-10-30, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2015-10-08 Created: 2015-09-14 Last updated: 2015-10-12Bibliographically approved

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Elfverson, DanielMålqvist, Axel

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