The completion of locally refined simplicial partitions created by bisection

Stevenson, Rob

doi:10.1090/S0025-5718-07-01959-X

The completion of locally refined simplicial partitions created by bisection
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by Rob Stevenson;

Math. Comp. 77 (2008), 227-241

DOI: https://doi.org/10.1090/S0025-5718-07-01959-X

Published electronically: July 26, 2007

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Abstract:

Recently, in [Found. Comput. Math., 7(2) (2007), 245–269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466–488] by Morin, Nochetto, and Siebert, converges with the optimal rate.The number of triangles $N$ in the output partition of such a method is generally larger than the number $M$ of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes.A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219–268] by Binev, Dahmen and DeVore saying that $N-N_0 \leq C M$ for some absolute constant $C$, where $N_0$ is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of $n$-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.

References

Douglas N. Arnold, Arup Mukherjee, and Luc Pouly, Locally adapted tetrahedral meshes using bisection, SIAM J. Sci. Comput. 22 (2000), no. 2, 431–448. MR 1780608, DOI 10.1137/S1064827597323373
Eberhard Bänsch, Local mesh refinement in $2$ and $3$ dimensions, Impact Comput. Sci. Engrg. 3 (1991), no. 3, 181–191. MR 1141298, DOI 10.1016/0899-8248(91)90006-G
Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
Jürgen Bey, Simplicial grid refinement: on Freudenthal’s algorithm and the optimal number of congruence classes, Numer. Math. 85 (2000), no. 1, 1–29. MR 1751367, DOI 10.1007/s002110050475
Igor Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288. MR 1329875, DOI 10.1016/0377-0427(94)90034-5
Joseph M. Maubach, Local bisection refinement for $n$-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995), no. 1, 210–227. MR 1311687, DOI 10.1137/0916014
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488. MR 1770058, DOI 10.1137/S0036142999360044
Alfred Schmidt and Kunibert G. Siebert, Design of adaptive finite element software, Lecture Notes in Computational Science and Engineering, vol. 42, Springer-Verlag, Berlin, 2005. The finite element toolbox ALBERTA; With 1 CD-ROM (Unix/Linux). MR 2127659
R.P. Stevenson. Optimality of a standard adaptive finite element method. Found. Comput. Math., 7(2):245–269, 2007.
C. T. Traxler, An algorithm for adaptive mesh refinement in $n$ dimensions, Computing 59 (1997), no. 2, 115–137. MR 1475530, DOI 10.1007/BF02684475