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Towards higher order immersed finite elements 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
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: urn:nbn:se:uu:diva-301937OAI: oai:DiVA.org:uu-301937DiVA: diva2:955664
Supervisors
Projects
eSSENCE
Available from: 2016-08-26 Created: 2016-08-25 Last updated: 2016-08-26Bibliographically approved
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, 364-387 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-298112 (URN)10.1016/j.cma.2016.06.001 (DOI)
External cooperation:
Projects
eSSENCE
Available from: 2016-06-21 Created: 2016-06-29 Last updated: 2016-08-26Bibliographically approved
2. Higher order cut elements for the wave equation
Open this publication in new window or tab >>Higher order cut elements for the wave equation
2016 (English)In: Computing Research Repository, no 1608.03107Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-301820 (URN)
External cooperation:
Projects
eSSENCE
Available from: 2016-08-10 Created: 2016-08-25 Last updated: 2016-08-26Bibliographically approved

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