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Global and local refinement techniques yielding nonobtuse tetrahedral partitions. (English) Zbl 1086.65116

A tetrahedron is said to be a path tetrahedron if its three edges, which do not meet at the same vertex are mutually orthogonal. The authors state a set of conditions on a tetrahedron \( T \) which ensures the existence of a family of partitions of \( T \) consisting of path tetrahedrons only [see M. Krizek and J. Pradlova, Numer. Methods Partial Differ. Equations 16, 327–334 (2000; Zbl 0957.65012)]. Considering a path tetrahedron \( ABCD \) such that the edges \( AB, BC \) and \( CD \) are mutually orthogonal, the authors prove that there exists an infinite family of nonobtuse partitions of it into path tetrahedra that locally refine \( ABCD \) in a neighbourhood of the vertex \( A \).

MSC:

65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35K15 Initial value problems for second-order parabolic equations

Citations:

Zbl 0957.65012
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References:

[1] Faragó, I.; Horváth, R., On the nonnegativity conservation of finite element solutions of parabolic problems, (), 78-86
[2] Korotov, S.; Kříek, M., Acute type refinements of tetrahedral partitions of polyhedral domains, SIAM J. numer. anal., 39, 724-733, (2001) · Zbl 1069.65017
[3] Vejchodský, T., Comparison principle for a nonlinear parabolic problem of a nonmonotone type, Appl. math., 29, 65-73, (2002) · Zbl 1014.35044
[4] Ciarlet, P.G.; Raviart, P.A., Maximum principle and uniform convergence for the finite element method, Comput. methods appl. mech. engrg., 2, 17-31, (1973) · Zbl 0251.65069
[5] Faragó, I., Qualitative properties of the numerical solution of linear parabolic problems with nonhomogeneous boundary conditions, Computers math. applic., 31, 4/5, 143-150, (1996) · Zbl 0874.65068
[6] Feistauer, M.; Felcman, J.; Rokyta, M.; Vlášek, Z., Finite-element solution of flow problems with trailing conditions, J. comput. appl. math., 44, 131-165, (1992) · Zbl 0766.76049
[7] Christie, I.; Hall, C., The maximum principle for bilinear elements, Internat. J. numer. methods engrg., 20, 549-553, (1984) · Zbl 0531.65058
[8] Fujii, H., Some remarks on finite element analysis of time-dependent field problems, (), 91-106
[9] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Tokyo · Zbl 0445.73043
[10] Bertollazzi, E., Discrete conservation and discrete maximum principle for elliptic pdes, Math. models methods appl. sci., 8, 685-711, (1998) · Zbl 0939.65123
[11] Korotov, S.; Kříek, M.; Neittaanmäki, P., Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle, Math. comp., 70, 107-119, (2001) · Zbl 1001.65125
[12] Stoyan, G., On a maximum principle for matrices, and on conservation of monotonicity. with applications to discretization methods, Z. angew. math. mech., 62, 375-381, (1982) · Zbl 0501.65011
[13] Stoyan, G., On maximum principles for monotone matrices, Linear algebra appl., 78, 147-161, (1986) · Zbl 0587.15014
[14] Kříek, M.; Qun, L., On diagonal dominance of stiffness matrices in 3D, East-west J. numer. math., 3, 59-69, (1995) · Zbl 0824.65112
[15] Kříek, M.; Neittaanmäki, P., Finite element approximation of variational problems and applications, (1990), Longman Scientific & Technical Amsterdam
[16] Kříek, M., An equilibrium finite element method in three-dimensional elasticity, Apt. mat., 27, 46-75, (1982)
[17] Zhang, S., Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes, Houston J. math., 21, 541-556, (1995) · Zbl 0855.65124
[18] Babuška, I.; Strouboulis, T., The finite element method and its reliability, (2001), Clarendon Press Harlow · Zbl 0997.74069
[19] Guo, B.Q., The h-p version of the finite element method for solving boundary value problems in polyhedral domains, (), 101-120, (Luminy, 1993)
[20] Bänsch, E., Local mesh refinement in 2 and 3 dimensions, Impact comput. sci. engrg., 3, 181-191, (1991) · Zbl 0744.65074
[21] Kříek, M.; Strouboulis, T., How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition, Numer. methods partial differential equations, 13, 201-214, (1997) · Zbl 0879.65078
[22] Golias, N.A.; Tsiboukis, T.D., An approach to refining three-dimensional tetrahedral meshes based on Delaunay transformations, Internat. J. numer. methods engrg., 37, 793-812, (1994) · Zbl 0796.73061
[23] Kříek, M.; Pradlová, J., Nonobtuse tetrahedral partitions, Numer. methods partial differential equations, 16, 327-334, (2000) · Zbl 0957.65012
[24] Korotov, S.; Kříek, M., Local nonobtuse tetrahedral refinements of a cube, Appl. math. lett., 16, 7, 1101-1104, (2003) · Zbl 1046.65106
[25] Gardner, M., Mathematical games, Scient. amer., 202, 172-186, (1960)
[26] Gerver, J.L., The dissection of a polygon into nearly equilateral triangles, Geom. dedicata, 16, 93-106, (1984) · Zbl 0547.05026
[27] Manheimer, W.; Federico, P.J., Dissecting an obtuse triangle into acute triangles, Amer. math. monthly, 67, 923, (1960)
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