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A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator
Department of Mathematics, University College London, 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, Umeå, Sweden.
Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
2020 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112803Article in journal (Refereed) Epub ahead of print
Abstract [en]

We consider solving the surface Helmholtz equation 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. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition hk<C, where h denotes the mesh size, k the wave number and C a constant depending mainly on the surface curvature κ, but not on the surface/mesh intersection. Optimal error estimates in the H1 and L2-norms follow.

Place, publisher, year, edition, pages
Elsevier, 2020. Vol. 362, article id 112803
Keywords [en]
Helmholtz-Beltrami, stabilization, TraceFEM, Finite element method, Galerkin methods, Mesh generation, Piecewise linear techniques, Beltrami, Discrete formulations, Galerkin Least Squares, Optimal error estimate, Surface curvatures, Three dimensional space, Two-dimensional surface, Least squares approximations
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:hj:diva-47450DOI: 10.1016/j.cma.2019.112803Scopus ID: 2-s2.0-85077774353Local ID: ;JTHMaterialISOAI: oai:DiVA.org:hj-47450DiVA, id: diva2:1386930
Available from: 2020-01-20 Created: 2020-01-20 Last updated: 2020-01-20

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Hansbo, Peter

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