×

zbMATH — the first resource for mathematics

Strongly regular family of boundary-fitted tetrahedral meshes of bounded \(C^2\) domains. (English) Zbl 1413.65424
Summary: We give a constructive proof that for any bounded domain of the class \(C^2\) there exists a strongly regular family of boundary-fitted tetrahedral meshes. We adopt a refinement technique introduced by M. Křížek [Apl. Mat. 27, 46–75 (1982; Zbl 0488.73072)] and modify it so that a refined mesh is again boundary-fitted. An alternative regularity criterion based on similarity with the Sommerville tetrahedron is used and shown to be equivalent to other standard criteria. The sequence of regularities during the refinement process is estimated from below and shown to converge to a positive number by virtue of the convergence of \(q\)-Pochhammer symbol. The final result takes the form of an implication with an assumption that can be obviously fulfilled for any bounded \(C^2\) domain.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] M. Audin: Geometry. Universitext, Springer, Berlin, 2003, 2012.
[2] Brandts, J.; Korotov, S.; Krížek, M., On the equivalence of ball conditions for simplicial finite elements in R\^{}{d}, Appl. Math. Lett., 22, 1210-1212, (2009) · Zbl 1173.52301
[3] Edelsbrunner, H., Triangulations and meshes in computational geometry, Acta Numerica, 9, 133-213, (2000) · Zbl 1004.65024
[4] L.C. Evans: Partial Differential Equations. Graduate Studies in Mathematics 19, American Mathematical Society, Providence, 1998. · Zbl 0902.35002
[5] E. Feireisl, R. Hošek, D. Maltese, A. Novotný: Error estimates for a numerical method for the compressible Navier-Stokes system on sufficiently smooth domains. To appear in ESAIM, Math. Model. Numer. Anal. (2016). Preprint IM-2015-46, available at http://math.cas.cz/fichier/preprints/IM_20150826112605_92.pdf, 2015.
[6] R. A. Horn, C.R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 2013. · Zbl 1267.15001
[7] Hošek, R., Face-to-face partition of 3D space with identical well-centered tetrahedra, Appl. Math., Praha, 60, 637-651, (2015) · Zbl 1363.65209
[8] Korotov, S.; Krížek, M.; Neittaanmäki, P., On the existence of strongly regular families of triangulations for domains with a piecewise smooth boundary, Appl. Math., Praha, 44, 33-42, (1999) · Zbl 1060.65665
[9] Křížek, M., An equilibrium finite element method in three-dimensional elasticity, Apl. Mat., 27, 46-75, (1982) · Zbl 0488.73072
[10] Sommerville, D.M.Y., Space-filling tetrahedra in Euclidean space, Proc. Edinburgh Math. Soc., 41, 49-57, (1923)
[11] Zhang, S., Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes, Houston J. Math., 21, 541-556, (1995) · Zbl 0855.65124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.