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A stabilized Nitsche cut element method for the wave equation
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
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.
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
Place, publisher, year, edition, pages
2016. Vol. 309, p. 364-387
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-298112DOI: 10.1016/j.cma.2016.06.001ISI: 000383828400015OAI: oai:DiVA.org:uu-298112DiVA, id: diva2:944567
Projects
eSSENCEAvailable from: 2016-06-21 Created: 2016-06-29 Last updated: 2018-04-10Bibliographically approved
In thesis
1. Towards higher order immersed finite elements for the wave equation
Open this publication in new window or tab >>Towards higher order immersed finite elements for the wave equation
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

We consider solving the scalar wave equation using immersed finite elements. Such a method might be useful, for instance, in scattering problems when the geometry of the domain is not known a priori. For hyperbolic problems, the amount of computational work per dispersion error is generally lower when using higher order methods. This serves as motivation for considering a higher order immersed method.

One problem in immersed methods is how to enforce boundary conditions. In the present work, boundary conditions are enforced weakly using Nitsche's method. This leads to a symmetric weak formulation, which is essential when solving the wave equation. Since the discrete system consists of symmetric matrices, having real eigenvalues, this ensures stability of the semi-discrete problem.

In immersed methods, small intersections between the immersed domain and the elements of the background mesh make the system ill-conditioned. This ill-conditioning becomes increasingly worse when using higher order elements. Here, we consider resolving this issue using additional stabilization terms. These terms consist of jumps in higher order derivatives acting on the internal faces of the elements intersected by the boundary.

Place, publisher, year, edition, pages
Uppsala University, 2016
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2016-008
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-301937 (URN)
External cooperation:
Supervisors
Projects
eSSENCE
Available from: 2016-08-26 Created: 2016-08-25 Last updated: 2016-08-26Bibliographically approved
2. High Order Cut Finite Element Methods for Wave Equations
Open this publication in new window or tab >>High Order Cut Finite Element Methods for Wave Equations
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
Cut finite element, Wave equation, Immersed, Fictitious domain
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-347439 (URN)978-91-513-0300-0 (ISBN)
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: 2018-05-04

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Sticko, SimonKreiss, Gunilla

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