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Full gradient stabilized cut finite element methods for surface partial differential equations

Burman, E; Hansbo, P; Larson, MG; Massing, A; Zahedi, S; (2016) Full gradient stabilized cut finite element methods for surface partial differential equations. Computer Methods in Applied Mechanics and Engineering , 310 (2016) pp. 278-296. 10.1016/j.cma.2016.06.033. Green open access

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Abstract

We propose and analyze a new stabilized cut finite element method for the Laplace–Beltrami operator on a closed surface. The new stabilization term provides control of the full R3 gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace–Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and L2 norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

Type: Article
Title: Full gradient stabilized cut finite element methods for surface partial differential equations
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.cma.2016.06.033
Publisher version: http://dx.doi.org/10.1016/j.cma.2016.06.033
Language: English
Additional information: Copyright © 2016. This manuscript version is published under a Creative Commons Attribution Non-commercial Non-derivative 4.0 International licence (CC BY-NC-ND 4.0). This licence allows you to share, copy, distribute and transmit the work for personal and non-commercial use providing author and publisher attribution is clearly stated. Further details about CC BY licences are available at http://creativecommons.org/licenses/by/4.0. Access may be initially restricted by the publisher.
Keywords: Surface PDE; Laplace–Beltrami operator; Cut finite element method; Stabilization; Condition number; A priori error estimates
UCL classification: UCL
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/1514228
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