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A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator
University College London.
Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
Umeå University.
2015 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, p. 188-207Article in journal (Refereed) Published
Abstract [en]

We consider solving the Laplace–Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.

The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method. 

Place, publisher, year, edition, pages
2015. Vol. 285, p. 188-207
Keywords [en]
Laplace–Beltrami; Embedded surface; Tangential calculus
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:hj:diva-25193DOI: 10.1016/j.cma.2014.10.044ISI: 000349637700009Scopus ID: 2-s2.0-84912115878Local ID: JTHProduktutvecklingISOAI: oai:DiVA.org:hj-25193DiVA, id: diva2:767049
Funder
Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708Swedish Foundation for Strategic Research , AM13-0029Available from: 2014-11-29 Created: 2014-11-29 Last updated: 2017-12-05Bibliographically approved

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Hansbo, Peter

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