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High Order Cut Finite Element Methods for Wave Equations
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: 0000-0002-4694-4731
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers wave propagation problems solved using finite element methods where a boundary or interface of the domain is not aligned with the computational mesh. Such methods are usually referred to as cut or immersed methods. The motivation for using immersed methods for wave propagation comes largely from scattering problems when the geometry of the domain is not known a priori. For wave propagation problems, the amount of computational work per dispersion error is generally lower when using a high order method. For this reason, this thesis aims at studying high order immersed methods.

Nitsche's method is a common way to assign boundary or interface conditions in immersed finite element methods. Here, penalty terms that are consistent with the boundary/interface conditions are added to the weak form. This requires that special quadrature rules are constructed on the intersected elements, which take the location of the immersed boundary/interface into account. A common problem for all immersed methods is small cuts occurring between the elements in the mesh and the computational domain. A suggested way to remedy this is to add terms penalizing jumps in normal derivatives over the faces of the intersected elements.

Paper I and Paper II consider the acoustic wave equation, using first order elements in Paper I, and using higher order elements in Paper II. High order elements are then used for the elastic wave equation in Paper III. Papers I to III all use continuous Galerkin, Nitsche's method, and jump-stabilization. Paper IV compares the errors of this type of cut finite element method with two other numerical methods. One result from Paper II is that the added jump-stabilization results in a mass matrix with a high condition number. This motivates the investigation of alternatives. Paper V considers a hybridizable discontinuous Galerkin method. This paper investigates to what extent local time stepping in combination with cell-merging can be used to overcome the problem of small cuts.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. , p. 37
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1656
Keywords [en]
Cut finite element, Wave equation, Immersed, Fictitious domain
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-347439ISBN: 978-91-513-0300-0 (print)OAI: oai:DiVA.org:uu-347439DiVA, id: diva2:1194410
Public defence
2018-05-25, ITC 2446, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2018-04-27 Created: 2018-04-02 Last updated: 2018-10-08
List of papers
1. A stabilized Nitsche cut element method for the wave equation
Open this publication in new window or tab >>A stabilized Nitsche cut element method for the wave equation
2016 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 309, p. 364-387Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-298112 (URN)10.1016/j.cma.2016.06.001 (DOI)000383828400015 ()
Projects
eSSENCE
Available from: 2016-06-21 Created: 2016-06-29 Last updated: 2018-04-10Bibliographically approved
2. Higher order cut finite elements for the wave equation
Open this publication in new window or tab >>Higher order cut finite elements for the wave equation
2019 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 80, p. 1867-1887Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-301820 (URN)10.1007/s10915-019-01004-2 (DOI)
Projects
eSSENCE
Available from: 2019-07-17 Created: 2016-08-25 Last updated: 2019-08-30Bibliographically approved
3. High order cut finite elements for the elastic wave equation
Open this publication in new window or tab >>High order cut finite elements for the elastic wave equation
2018 (English)In: Computing Research Repository, no 1804.00332Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-347421 (URN)
Projects
eSSENCE
Available from: 2018-04-01 Created: 2018-04-01 Last updated: 2018-04-10Bibliographically approved
4. High-order numerical methods for 2D parabolic problems in single and composite domains
Open this publication in new window or tab >>High-order numerical methods for 2D parabolic problems in single and composite domains
Show others...
2018 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 76, p. 812-847Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-339130 (URN)10.1007/s10915-017-0637-y (DOI)000436253800006 ()
Available from: 2018-01-10 Created: 2018-01-16 Last updated: 2018-09-09Bibliographically approved
5. High order cut discontinuous Galerkin methods with local time stepping for acoustics
Open this publication in new window or tab >>High order cut discontinuous Galerkin methods with local time stepping for acoustics
2018 (English)In: Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-347423 (URN)
Projects
eSSENCE
Available from: 2018-04-01 Created: 2018-04-01 Last updated: 2018-04-10Bibliographically approved

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