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Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions
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.
2019 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, no 6, p. 2247-2282Article in journal (Refereed) Published
Abstract [en]

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in R-d of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R-3.

Place, publisher, year, edition, pages
EDP Sciences, 2019. Vol. 52, no 6, p. 2247-2282
Keywords [en]
Surface PDE, Laplace-Beltrami operator, cut finite element method, stabilization, condition number, a priori error estimates, arbitrary codimension
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-162515DOI: 10.1051/m2an/2018038ISI: 000457984700005OAI: oai:DiVA.org:umu-162515DiVA, id: diva2:1344564
Available from: 2019-08-21 Created: 2019-08-21 Last updated: 2019-08-21Bibliographically approved

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Larson, Mats G.Massing, André

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  • de-DE
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