Volume 5, Issue 1
Numerical Characterization of the Regularity Loss in Minimal Surfaces

M. Hamouda & H. V. J. Le Meur

Int. J. Numer. Anal. Mod., 5 (2008), pp. 152-166

Published online: 2008-05

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  • Abstract

In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the data. Firstly, we validate the numerical study on the astroid and discuss the various kinds of non-regularity characterizations. We provide an algorithm to test the regularity loss using the numerical results. Secondly, we give a numerical estimate of the threshold value of the boundary condition beyond which no regular solution exists. More theoretical results are also given on the approximation by the regularized solution of the non-regularized one. The regularized solution exhibits a boundary layer. Finally, the study is applied to the catenoid for which the exact threshold value is known. The exact value and the computed one are in good agreement.

  • Keywords

boundary layers minimal surfaces non-regular solutions singular perturbations

  • AMS Subject Headings

65N22 35J25 35B30 35B65 49M25 35D10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-5-152, author = {M. Hamouda and H. V. J. Le Meur}, title = {Numerical Characterization of the Regularity Loss in Minimal Surfaces}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {1}, pages = {152--166}, abstract = {In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the data. Firstly, we validate the numerical study on the astroid and discuss the various kinds of non-regularity characterizations. We provide an algorithm to test the regularity loss using the numerical results. Secondly, we give a numerical estimate of the threshold value of the boundary condition beyond which no regular solution exists. More theoretical results are also given on the approximation by the regularized solution of the non-regularized one. The regularized solution exhibits a boundary layer. Finally, the study is applied to the catenoid for which the exact threshold value is known. The exact value and the computed one are in good agreement. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/804.html} }
TY - JOUR T1 - Numerical Characterization of the Regularity Loss in Minimal Surfaces AU - M. Hamouda & H. V. J. Le Meur JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 152 EP - 166 PY - 2008 DA - 2008/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/804.html KW - boundary layers KW - minimal surfaces KW - non-regular solutions KW - singular perturbations AB - In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the data. Firstly, we validate the numerical study on the astroid and discuss the various kinds of non-regularity characterizations. We provide an algorithm to test the regularity loss using the numerical results. Secondly, we give a numerical estimate of the threshold value of the boundary condition beyond which no regular solution exists. More theoretical results are also given on the approximation by the regularized solution of the non-regularized one. The regularized solution exhibits a boundary layer. Finally, the study is applied to the catenoid for which the exact threshold value is known. The exact value and the computed one are in good agreement.
M. Hamouda & H. V. J. Le Meur. (1970). Numerical Characterization of the Regularity Loss in Minimal Surfaces. International Journal of Numerical Analysis and Modeling. 5 (1). 152-166. doi:
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