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A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator

Burman, E; Hansbo, P; Larson, MG; Massing, A; (2020) A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator. Computer Methods in Applied Mechanics and Engineering , 362 , Article 112803. 10.1016/j.cma.2019.112803. (In press). Green open access

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Abstract

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.

Type: Article
Title: A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.cma.2019.112803
Publisher version: https://doi.org/10.1016/j.cma.2019.112803
Language: English
Additional information: Helmholtz-Beltrami, TraceFEM, stabilization
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10090691
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