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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.
Högskolan i Jönköping, Tekniska Högskolan, JTH, Material och tillverkning.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 (Engelska)Ingår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, nr 6, s. 2247-2282Artikel i tidskrift (Refereegranskat) 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.

Ort, förlag, år, upplaga, sidor
EDP Sciences, 2019. Vol. 52, nr 6, s. 2247-2282
Nyckelord [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
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:hj:diva-43224DOI: 10.1051/m2an/2018038ISI: 000457984700005Scopus ID: 2-s2.0-85052126397Lokalt ID: ;JTHMaterialISOAI: oai:DiVA.org:hj-43224DiVA, id: diva2:1292986
Tillgänglig från: 2019-03-01 Skapad: 2019-03-01 Senast uppdaterad: 2019-03-01Bibliografiskt granskad

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