Ninth International Conference Zaragoza-Pau on Applied Mathematics and Statistics Jaca, September 19–21th 2005
Geometric and Functional Aspects of Some Triangulations of Sphere–like Bodies Natalia BOAL1 , V´ıctor DOM´ INGUEZ2 , Francisco-Javier SAYAS1 SUMMARY In this talk we will deal with the Baumgardner and Frederickson icosahedral triangulation of the sphere [1] and some extensions. This family of triangulations is defined by midpoint subdivision of an initial one and has several interesting properties from the computational point of view. In a previous paper [2], we had observed some good qualities of this triangulation as a means to construct an automatic quadrature method on spherical polygons. We are now able to prove some conjectures that where stated there and that justify the process. The main property is related to the fact that the inscribed polyhedron related to each level of the discretization process can be mapped back and forth to the sphere with a uniformly Lipschitz mapping. This fact proves quasi–uniformity of the triangulation and shows that the basic Sobolev spaces (of fractional orders between −1 and 1) on the sphere and on the set of polyhedra are isomorphic with constants independent of the discretization parameter. We will finally hint at how this result can be used to begin a mathematical understanding of the way that the boundary element method is used by engineers. Keywords:
Triangulations, sphere, Sobolev spaces
AMS Classification:
65D17, 65D32, 65N38
References [1] J. R. Baumgardner and P. O. Frederickson. Icosahedral discretization of the two-sphere. SIAM J. Numer. Anal 22(6), 1107–1115, 1985. [2] N. Boal and F.-J. Sayas. Adaptive numerical integration on spherical triangles. In VIII Journ´ees Zaragoza-Pau de Math´ematiques Appliqu´ees et de Statistiques, Monogr. Semin. Mat. Garc´ıa Galdeano 31, M. C. L´opez de Silanes et al. (eds.), pp. 61–69. Universidad de Zaragoza, 2004.
1
Departmento de Matem´ atica Aplicada Universidad de Zaragoza Centro Polit´ecnico Superior, 50018 Zaragoza
[email protected],
[email protected]
2
Departmento de Matem´ atica e Inform´atica Universidad P´ ublica de Navarra Campus de Arrosad´ıa, Pamplona
[email protected]
