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A stabilized cut finite element method for the Darcy problem on surfaces
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 298-318Article in journal (Refereed) Published
Abstract [en]

We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three dimensional mesh as trial and test functions. Since we consider a partial differential equation on a surface, the resulting discrete weak problem might be severely ill conditioned. We propose a full gradient and a normal gradient based stabilization computed on the background mesh to render the proposed formulation stable and well conditioned irrespective of the surface positioning within the mesh. Our formulation extends and simplifies the Masud-Hughes stabilized primal mixed formulation of the Darcy surface problem proposed in Hansbo and Larson (2016) on fitted triangulated surfaces. The tangential condition on the velocity and the pressure gradient is enforced only weakly, avoiding the need for any tangential projection. The presented numerical analysis accounts for different polynomial orders for the velocity, pressure, and geometry approximation which are corroborated by numerical experiments. In particular, we demonstrate both theoretically and through numerical results that the normal gradient stabilized variant results in a high order scheme.

Place, publisher, year, edition, pages
Elsevier, 2017. Vol. 326, p. 298-318
Keyword [en]
Surface PDE, Darcy problem, Cut finite element method, Stabilization, Condition number, A priori error estimates
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-141801DOI: 10.1016/j.cma.2017.08.007ISI: 000413322300013OAI: oai:DiVA.org:umu-141801DiVA, id: diva2:1161120
Available from: 2017-11-29 Created: 2017-11-29 Last updated: 2018-06-09Bibliographically approved

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Larson, Mats G.Massing, André

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CiteExportLink to record
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Citation style
  • apa
  • ieee
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  • Other style
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  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
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More languages
Output format
  • html
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  • asciidoc
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