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Shape calculus and finite element method in smooth domains


Author: T. Tiihonen
Journal: Math. Comp. 70 (2001), 1-15
MSC (2000): Primary 65N30; Secondary 49Q12
DOI: https://doi.org/10.1090/S0025-5718-00-01323-5
Published electronically: October 2, 2000
MathSciNet review: 1803123
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Abstract: The use of finite elements in smooth domains leads naturally to polyhedral or piecewise polynomial approximations of the boundary. Hence the approximation error consists of two parts: the geometric part and the finite element part. We propose to exploit this decomposition in the error analysis by introducing an auxiliary problem defined in a polygonal domain approximating the original smooth domain. The finite element part of the error can be treated in the standard way. To estimate the geometric part of the error, we need quantitative estimates related to perturbation of the geometry. We derive such estimates using the techniques developed for shape sensitivity analysis.


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Additional Information

T. Tiihonen
Affiliation: Department of Mathematical Information Technology, University of Jyväskylä, Box 35, FIN–40351 Jyväskylä, Finland
Email: tiihonen@mit.jyu.fi

DOI: https://doi.org/10.1090/S0025-5718-00-01323-5
Keywords: Finite elements, curved boundary, error estimates, shape derivatives, continuous dependence on geometry
Received by editor(s): November 17, 1997
Published electronically: October 2, 2000
Article copyright: © Copyright 2000 American Mathematical Society