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Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions
Department of Mathematics, University College London, United Kingdom.
Jönköping University, School of Engineering, JTH, Materials and Manufacturing.ORCID iD: 0000-0001-7352-1550
Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
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 Rd 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 R3.

Place, publisher, year, edition, pages
EDP Sciences, 2019. Vol. 52, no 6, p. 2247-2282
Keywords [en]
A priori error estimates, Arbitrary codimension, Condition number, Cut finite element method, Laplace-Beltrami operator, Stabilization, Surface PDE, Embeddings, Laplace transforms, Mesh generation, Number theory, Piecewise linear techniques, Stiffness matrix, A-priori estimates, Codimension, Condition number of the stiffness matrix, Condition numbers, Numerical results, Priori error estimate, Theoretical framework, Finite element method
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:hj:diva-43224DOI: 10.1051/m2an/2018038ISI: 000457984700005Scopus ID: 2-s2.0-85052126397Local ID: ;JTHMaterialISOAI: oai:DiVA.org:hj-43224DiVA, id: diva2:1292986
Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2019-03-01Bibliographically approved

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Hansbo, Peter

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