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A cut finite element method with boundary value correction


Authors: Erik Burman, Peter Hansbo and Mats G. Larson
Journal: Math. Comp. 87 (2018), 633-657
MSC (2010): Primary 65M60; Secondary 65M85
DOI: https://doi.org/10.1090/mcom/3240
Published electronically: October 11, 2017
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Abstract: In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomée in [Math. Comp. 26 (1972), 869-879]. The cut finite element method is a fictitious domain method with Nitsche-type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.


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Additional Information

Erik Burman
Affiliation: Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom

Peter Hansbo
Affiliation: Department of Mechanical Engineering, Jönköping University, S-551 11 Jönköping, Sweden

Mats G. Larson
Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden
Email: mats.larson@math.umu.se

DOI: https://doi.org/10.1090/mcom/3240
Received by editor(s): July 11, 2015
Received by editor(s) in revised form: July 8, 2016, and October 31, 2016
Published electronically: October 11, 2017
Additional Notes: This research was supported in part by EPSRC, UK, Grant No. EP/J002313/1, the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grants Nos. 2011-4992, 2013-4708, and Swedish strategic research programme eSSENCE
Article copyright: © Copyright 2017 American Mathematical Society

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