| Title:
|
Nonobtuse tetrahedral partitions that refine locally towards Fichera-like corners (English) |
| Author:
|
Beilina, Larisa |
| Author:
|
Korotov, Sergey |
| Author:
|
Křížek, Michal |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
50 |
| Issue:
|
6 |
| Year:
|
2005 |
| Pages:
|
569-581 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain. (English) |
| Keyword:
|
partial differential equations |
| Keyword:
|
finite element method |
| Keyword:
|
path tetrahedron |
| Keyword:
|
linear tetrahedral finite element |
| Keyword:
|
discrete maximum principle |
| Keyword:
|
reentrant corner |
| Keyword:
|
Fichera vertex |
| Keyword:
|
nonlinear heat conduction |
| MSC:
|
51M20 |
| MSC:
|
65N30 |
| MSC:
|
65N50 |
| idZBL:
|
Zbl 1099.65105 |
| idMR:
|
MR2181027 |
| DOI:
|
10.1007/s10492-005-0038-7 |
| . |
| Date available:
|
2009-09-22T18:24:17Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134624 |
| . |
| Reference:
|
[1] T. Apel: Anisotropic Finite Elements: Local Estimates and Applications. Advances in Numer. Math.B. G. Teubner, Leipzig, 1999. MR 1716824 |
| Reference:
|
[2] T. Apel, F. Milde: Comparison of several mesh refinement strategies near edges.Commun. Numer. Methods Eng. 12 (1996), 373–381. 10.1002/(SICI)1099-0887(199607)12:7<373::AID-CNM985>3.0.CO;2-8 |
| Reference:
|
[3] T. Apel, S. Nicaise: The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges.Math. Methods Appl. Sci. 21 (1998), 519–549. MR 1615426, 10.1002/(SICI)1099-1476(199804)21:6<519::AID-MMA962>3.0.CO;2-R |
| Reference:
|
[4] O. Axelsson, V. A. Barker: Finite Element Solution of Boundary Value Problems. Theory and Computation.Academic Press, Orlando, 1984. MR 0758437 |
| Reference:
|
[5] M. Bern, P. Chew, D. Eppstein, and J. Ruppert: Dihedral bounds for mesh generation in high dimensions.Proc. of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1995), SIAM, Philadelphia, 1995, pp. 189–196. MR 1321850 |
| Reference:
|
[6] F. Bornemann, B. Erdmann, and R. Kornhuber: Adaptive multilevel methods in three space dimensions.Int. J. Numer. Methods Eng. 36 (1993), 3187–3203. MR 1236370, 10.1002/nme.1620361808 |
| Reference:
|
[7] E. Bänsch: Local mesh refinement in 2 and 3 dimensions.IMPACT Comput. Sci. Eng. 3 (1991), 181–191. MR 1141298, 10.1016/0899-8248(91)90006-G |
| Reference:
|
[8] L. Collatz: Numerische Behandlung von Differentialgleichungen.Springer-Verlag, Berlin-Göttingen-Heidelberg, 1951. Zbl 0054.05101, MR 0043563 |
| Reference:
|
[9] R. Cools: Monomial cubature rules since “Stroud”: A compilation. II.J. Comput. Appl. Math. 112 (1999), 21–27. Zbl 0954.65021, MR 1728449, 10.1016/S0377-0427(99)00229-0 |
| Reference:
|
[10] R. Cools, P. Rabinowitz: Monomial cubature rules since “Stroud”: A compilation.J. Comput. Appl. Math. 48 (1993), 309–326. MR 1252544, 10.1016/0377-0427(93)90027-9 |
| Reference:
|
[11] H. S. M. Coxeter: Trisecting an orthoscheme.Comput. Math. Appl. 17 (1989), 59–71. Zbl 0706.51019, MR 0994189, 10.1016/0898-1221(89)90148-X |
| Reference:
|
[12] M. Dauge: Elliptic Boundary Value Problems on Corner Domains. Lect. Notes Math., Vol. 1341.Springer-Verlag, Berlin, 1988. MR 0961439 |
| Reference:
|
[13] M. Feistauer, J. Felcman, M. Rokyta, and Z. Vlášek: Finite-element solution of flow problems with trailing conditions.J. Comput. Appl. Math. 44 (1992), 131–165. MR 1197680, 10.1016/0377-0427(92)90008-L |
| Reference:
|
[14] G. Fichera: Numerical and Quantitative Analysis. Surveys and Reference Works in Mathematics, Vol. 3.Pitman (Advanced Publishing Program), London-San Francisco-Melbourne, 1978. MR 0519677 |
| Reference:
|
[15] H. Fujii: Some remarks on finite element analysis of time-dependent field problems.Theory Pract. Finite Elem. Struct. Analysis, Univ. Tokyo Press, Tokyo, 1973, pp. 91–106. Zbl 0373.65047 |
| Reference:
|
[16] B. Q. Guo: The $h$-$p$ version of the finite element method for solving boundary value problems in polyhedral domains.Boundary Value Problems and Integral Equations in Nonsmooth Domains (Luminy, 1993). Lect. Notes Pure Appl. Math., Vol. 167, M. Costabel, M. Dauge, and C. Nicaise (eds.), Marcel Dekker, New York, 1995, pp. 101–120. Zbl 0855.65114, MR 1301344 |
| Reference:
|
[17] P. Keast: Moderate-degree tetrahedral quadrature formulas.Comput. Methods Appl. Mech. Eng. 55 (1986), 339–348. Zbl 0572.65008, MR 0844909, 10.1016/0045-7825(86)90059-9 |
| Reference:
|
[18] S. Korotov, M. Křížek: Acute type refinements of tetrahedral partitions of polyhedral domains.SIAM J. Numer. Anal. 39 (2001), 724–733. MR 1860255, 10.1137/S003614290037040X |
| Reference:
|
[19] S. Korotov, M. Křížek: Local nonobtuse tetrahedral refinements of a cube.Appl. Math. Lett. 16 (2003), 1101–1104. MR 2013079, 10.1016/S0893-9659(03)90101-7 |
| Reference:
|
[20] I. Kossaczký: A recursive approach to local mesh refinement in two and three dimensions.J. Comput. Appl. Math. 55 (1994), 275–288. MR 1329875, 10.1016/0377-0427(94)90034-5 |
| Reference:
|
[21] M. Křížek, L. Liu: On the maximum and comparison principles for a steady-state nonlinear heat conduction problem.Z. Angew. Math. Mech. 83 (2003), 559–563. MR 1994036, 10.1002/zamm.200310054 |
| Reference:
|
[22] M. Křížek, Qun Lin: On diagonal dominance of stiffness matrices in 3D.East-West J. Numer. Math. 3 (1995), 59–69. MR 1331484 |
| Reference:
|
[23] M. Křížek, T. Strouboulis: How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition.Numer. Methods Partial Differ. Equations 13 (1997), 201–214. MR 1436615, 10.1002/(SICI)1098-2426(199703)13:2<201::AID-NUM5>3.0.CO;2-T |
| Reference:
|
[24] J. M. Maubach: Local bisection refinement for $n$-simplicial grids generated by reflection.SIAM J. Sci. Comput. 16 (1995), 210–227. Zbl 0816.65090, MR 1311687, 10.1137/0916014 |
| Reference:
|
[25] M. Picasso: Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz-Zhu error estimator.Commun. Numer. Methods Eng. 19 (2003), 13–23. Zbl 1021.65052, MR 1952014, 10.1002/cnm.546 |
| Reference:
|
[26] M. Picasso: An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: Application to elliptic and parabolic problems.SIAM J. Sci. Comput. 24 (2003), 1328–1355. Zbl 1061.65116, MR 1976219, 10.1137/S1064827501398578 |
| Reference:
|
[27] A. Plaza, G. F. Carey: Local refinement of simplicial grids based on the skeleton.Appl. Numer. Math. 32 (2000), 195–218. MR 1734507, 10.1016/S0168-9274(99)00022-7 |
| Reference:
|
[28] H. Schmitz, K. Volk, and W. Wendland: Three-dimensional singularities of elastic fields near vertices.Numer. Methods Partial Differ. Equations 9 (1993), 323–337. MR 1216118, 10.1002/num.1690090309 |
| Reference:
|
[29] R. S. Varga: Matrix iterative analysis.Prentice-Hall, New Jersey, 1962. MR 0158502 |
| . |
