Year: 2016
Author: Mahdieh Sattari, Jukka Tuomela
Journal of Computational Mathematics, Vol. 34 (2016), Iss. 1 : pp. 12–29
Abstract
Some models dealing with fibers and liquid crystals can be formulated probabilistically in terms of orientation distributions. Since the orientation of a thin object can be specified by a point in a real projective plane this approach leads to elliptic and parabolic problems in the real projective plane. In most previous works these kinds of problems have been considered on the unit sphere which is a double cover of the real projective plane. However, numerically this is inefficient because the resulting systems of equations are unnecessarily big. We formulate the problem directly in the real projective plane using a certain parametrization with three coordinate domains. After reducing the computations to the coordinate domains we can then use finite elements almost in a standard way. In particular the standard error estimates with usual Sobolev spaces remain valid in this setting. We consider both elliptic and parabolic cases, and demonstrate the validity of our approach.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1510-m4504
Journal of Computational Mathematics, Vol. 34 (2016), Iss. 1 : pp. 12–29
Published online: 2016-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: Smoluchowski equation Real projective plane Finite elements Orientation distribution.
Author Details
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Simulation of the spherical orientation probability distribution of paper fibers in an entire suspension using immersed boundary methods
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Röyttä, Pekka
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https://doi.org/10.1016/j.jnnfm.2016.01.001 [Citations: 10]