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Volume 34, Issue 1
On the Numerical Solution of Elliptic and Parabolic PDE in the Real Projective Plane

Mahdieh Sattari & Jukka Tuomela

DOI: 10.4208/jcm.1510-m4504

J. Comp. Math., 34 (2016), pp. 12-29.

Published online: 2016-02

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  • 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.

  • Keywords

Smoluchowski equation, Real projective plane, Finite elements, Orientation distribution.

  • AMS Subject Headings

65M60, 35Q84, 35Q35, 58J32.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mahdieh.sattari@uef.fi (Mahdieh Sattari)

jukka.tuomela@uef.fi (Jukka Tuomela)

  • BibTex
  • RIS
  • TXT
@Article{JCM-34-12, author = {Sattari , Mahdieh and Tuomela , Jukka}, title = {On the Numerical Solution of Elliptic and Parabolic PDE in the Real Projective Plane}, journal = {Journal of Computational Mathematics}, year = {2016}, volume = {34}, number = {1}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1510-m4504}, url = {http://global-sci.org/intro/article_detail/jcm/9780.html} }
TY - JOUR T1 - On the Numerical Solution of Elliptic and Parabolic PDE in the Real Projective Plane AU - Sattari , Mahdieh AU - Tuomela , Jukka JO - Journal of Computational Mathematics VL - 1 SP - 12 EP - 29 PY - 2016 DA - 2016/02 SN - 34 DO - http://doi.org/10.4208/jcm.1510-m4504 UR - https://global-sci.org/intro/article_detail/jcm/9780.html KW - Smoluchowski equation, Real projective plane, Finite elements, Orientation distribution. AB -

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.

Sattari , Mahdieh and Tuomela , Jukka. (2016). On the Numerical Solution of Elliptic and Parabolic PDE in the Real Projective Plane. Journal of Computational Mathematics. 34 (1). 12-29. doi:10.4208/jcm.1510-m4504
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