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Higher order cut finite elements for the wave equation
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.ORCID-id: 0000-0002-4694-4731
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.ORCID-id: 0000-0001-8865-8218
2019 (Engelska)Ingår i: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 80, s. 1867-1887Artikel i tidskrift (Refereegranskat) Published
Ort, förlag, år, upplaga, sidor
2019. Vol. 80, s. 1867-1887
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:uu:diva-301820DOI: 10.1007/s10915-019-01004-2ISI: 000482484200022OAI: oai:DiVA.org:uu-301820DiVA, id: diva2:955501
Projekt
eSSENCETillgänglig från: 2019-07-17 Skapad: 2016-08-25 Senast uppdaterad: 2019-10-29Bibliografiskt granskad
Ingår i avhandling
1. Towards higher order immersed finite elements for the wave equation
Öppna denna publikation i ny flik eller fönster >>Towards higher order immersed finite elements for the wave equation
2016 (Engelska)Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala University, 2016
Serie
IT licentiate theses / Uppsala University, Department of Information Technology, ISSN 1404-5117 ; 2016-008
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
urn:nbn:se:uu:diva-301937 (URN)
Externt samarbete:
Handledare
Projekt
eSSENCE
Tillgänglig från: 2016-08-26 Skapad: 2016-08-25 Senast uppdaterad: 2016-08-26Bibliografiskt granskad
2. High Order Cut Finite Element Methods for Wave Equations
Öppna denna publikation i ny flik eller fönster >>High Order Cut Finite Element Methods for Wave Equations
2018 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala: Acta Universitatis Upsaliensis, 2018. s. 37
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1656
Nyckelord
Cut finite element, Wave equation, Immersed, Fictitious domain
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Beräkningsvetenskap med inriktning mot numerisk analys
Identifikatorer
urn:nbn:se:uu:diva-347439 (URN)978-91-513-0300-0 (ISBN)
Disputation
2018-05-25, ITC 2446, Lägerhyddsvägen 2, Uppsala, 10:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2018-04-27 Skapad: 2018-04-02 Senast uppdaterad: 2018-10-08

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