Geometric separators for finite-element meshes

SIAM Journal on Scientific Computing

Volumn 19, Issue 2, 1998, Pages 364-386

  Miller, Gary L ^a   Teng, Shang Hua ^b   Thurston, William ^c   Vavasis, Stephen A ^d  

^a Carnegie Mellon University   (United States)

^b University of Minnesota   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d Department of Computer Science and School of Operations Research and Information Engineering   (United States)

Author keywords

Center points; Computational geometry; Conformal mapping; Domain decomposition; Finite elements; Graph separators; Mesh partitioning; Sparse matrix computations

Indexed keywords

EID: 0004505622     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827594262613     Document Type: Article

References (50)

1. N. ALON, P. SEYMOUR, AND R. THOMAS, A separator theorem for graphs with an excluded minor and its applications, in Proc. of the 22th Annual ACM Symposium on Theory of Computing, ACM, New York, 1990, pp. 293-299.
2. I. BABUŠKA AND A. K. AZIZ, On the angle condition in the finite element method, SIAM J. Numer. Anal., 13 (1976), pp. 214-226.
3. M. J. BERGER AND S. BOKHARI, A partitioning strategy for nonuniform problems on multi-processors, IEEE Trans. Comput., C-36 (1987), pp. 570-580.
4. M. BERN, D. EPPSTEIN, AND J. R. GILBERT, Provably good mesh generation, in Proc. 31st Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1990, pp. 231-241.
5. J. H. BRAMBLE, J. E. PASCIAK, AND A. H. SCHATZ, An iterative method for elliptic problems on regions partitioned into substructures, Math. Comp., 46 (1986), pp. 361-9.
6. L. P. CHEW, Guaranteed Quality Triangular Meshes, Tech. report 89-893, Department of Computer Science, Cornell University, Ithaca, NY, 1989.
7. K. CLARKSON, D. EPPSTEIN, G. L. MILLER, C. STURTIVANT, AND S.-H. TENG, Approximating center points with iterated radon points, Internat. J. Comput. Geom. Appl., 6 (1996), pp. 357-377.
8. L. DANZER, J. FONLUPT, AND V. KLEE, Helly's theorem and its relatives, in Proc. of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, 7 (1963), pp. 101-180.
9. A. L. DULMAGE AND N. S. MENDELSOHN, Coverings of bipartite graphs, Canadian J. Math., 10 (1958), pp. 517-534.
10. D. EPPSTEIN, G. L. MILLER, AND S.-H. TENG, A deterministic linear time algorithm for geometric separators and its applications, Fund. Inform., 22 (1995), pp. 309-330.
11. I. FRIED, Condition of finite element matrices generated from nonuniform meshes, AIAA J., 10 (1972), pp. 219-221.
12. J. E. L. G. HARDY AND G. PÓLYA, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952.
13. J. A. GEORGE, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345-363.
14. J. A. GEORGE AND J. W. H. LIU, An automatic nested dissection algorithm for irregular finite element problems, SIAM J. Numer. Anal., 15 (1978), pp. 1053-1069.
15. J. GILBERT, J. HUTCHINSON, AND R. TARJAN, A separation theorem for graphs of bounded genus, J. Algorithms, 5 (1984), pp. 391-407.
16. J. GILBERT, G. MILLER, AND S.-H. TENG, Geometric mesh partitioning: implementation and experiments, in Proc. of the 9th International Parallel Processing Symposium, IEEE, Piscataway, NJ, 1995, pp. 418-427; also Tech. report CSL-94-13, Xerox Palo Alto Research Center, Palo Alto, CA; SIAM J. Sci. Comput., to appear.
17. J. GILBERT, G. MILLER, AND S.-H. TENG, Geometric mesh partitioning: implementation and experiments, in Proc. of the 9th International Parallel Processing Symposium, IEEE, Piscataway, NJ, 1995, pp. 418-427; also Tech. report CSL-94-13, Xerox Palo Alto Research Center, Palo Alto, CA; SIAM J. Sci. Comput., to appear.
18. J. GILBERT, G. MILLER, AND S.-H. TENG, Geometric mesh partitioning: implementation and experiments, in Proc. of the 9th International Parallel Processing Symposium, IEEE, Piscataway, NJ, 1995, pp. 418-427; also Tech. report CSL-94-13, Xerox Palo Alto Research Center, Palo Alto, CA; SIAM J. Sci. Comput., to appear.
19. G. H. GOLUB AND C. F. V. LOAN, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, MD, 1989.
20. S. GUATTERY AND G. L. MILLER, On spectral partitioning methods, in Proc. of the 6th ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 1995, pp. 233-242.
21. D. HAUSSLER AND E. WELZL, ∈-net and simplex range queries, Discrete Comput. Geom., 2 (1987), pp. 127-151.
22. B. HENDRICKSON AND R. LELAND, An Improved Spectral Graph Partitioning Algorithm for Mapping Parallel Computations, SIAM J. Sci. Comput., 16 (1995), pp. 452-469.
23. B. HENDRICKSON AND R. LELAND, A multilevel algorithm for partitioning graphs, in Proc. Supercomputing '95, ACM, New York, 1995.
24. C. JOHNSON, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987.
25. B. KERNIGHAN AND S. LIN, An efficient heuristic procedure for partitioning graphs, Bell System Technical Journal, 29 (1970), pp. 291-307.
26. F. T. LEIGHTON AND S. RAO, An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms, in Proc. 29th Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1988, pp. 422-431.
27. R. J. LIPTON, D. J. ROSE, AND R. E. TARJAN, Generalized nested dissection, SIAM J. Numer. Anal., 16 (1979), pp. 346-358.
28. R. J. LIPTON AND R. E. TARJAN, A separator theorem for planar graphs, SIAM J. Appl. Math., 36 (1979), pp. 177-189.
29. J. W. H. LIU, The multifrontal method for sparse matrix solution: theory and practice, SIAM Rev., 34 (1992), pp. 82-109.
30. J. MATOUŠEK, Approximations and optimal geometric divide-and-conquer, in Proc. 23rd ACM Symposium Theory of Computing, ACM, New York, 1991, pp. 512-522.
31. G. L. MILLER, D. TALMOR, S.-H. TENG, AND N. WALKINGTON, A Delaunay based numerical method for three dimensions: generation, formulation, and partition, in Proc. of the 27th Annual ACM Symposium on the Theory of Computing, ACM, New York, 1995, pp. 683-692.
32. G. L. MILLER, S.-H. TENG, W. THURSTON, AND S. A. VAVASIS, Automatic mesh partitioning, in Graph Theory and Sparse Matrix Computation, A. George, J. Gilbert, and J. Liu, eds., Vol. 56 of IMA Vol. Math. Appl., Springer, New York, 1993, pp. 57-84.
33. G. L. MILLER, S.-H. TENG, W. THURSTON, AND S. A. VAVASIS, Separators for sphere-packings and nearest neighborhood graphs, J. ACM, 44 (1997), pp. 1-29.
34. G. L. MILLER, S.-H. TENG, W. THURSTON, AND S. A. VAVASIS, Geometric Separators for Finite Element Meshes: Appendix Concerning Volume Elements, 1994, unpublished; available via ftp://ftp.cs.cornell.edu/pub/vavasis/papers/diffgeo.ps.
35. G. L. MILLER, S.-H. TENG, AND S. A. VAVASIS, A unified geometric approach to graph separators, in Proc. 31nd Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1991, pp. 538-547.
36. G. L. MILLER AND W. THURSTON, Separators in two and three dimensions, in Proc. of the 22th Annual ACM Symposium on Theory of Computing, Baltimore, May 1990, ACM, New York, pp. 300-309.
37. G. L. MILLER AND S. A. VAVASIS, Density graphs and separators, in Proc. of the ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 1991, pp. 331-336.
38. S. A. MITCHELL AND S. A. VAVASIS, Quality mesh generation in three dimensions, in Proc. of the ACM Computational Geometry Conference, 1992, ACM, New York, pp. 212-221; also Tech. report C.S. TR 92-1267, Cornell University, Ithaca, NY.
39. S. A. MITCHELL AND S. A. VAVASIS, Quality mesh generation in three dimensions, in Proc. of the ACM Computational Geometry Conference, 1992, ACM, New York, pp. 212-221; also Tech. report C.S. TR 92-1267, Cornell University, Ithaca, NY.
40. V. PAN AND J. REIF, Efficient parallel solution of linear systems, in Proc. of the 17th Annual ACM Symposium on Theory of Computing, ACM, New York, 1985, pp. 143-152.
41. C. H. PAPADIMITRIOU AND K. STEIGLITZ, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ, 1982.
42. A. POTHEN, H. D. SIMON, AND K.-P. LIOU, Partitioning sparse matrices with eigenvectors of graphs, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 430-452.
43. J. RUPPERT, A new and simple algorithm for quality 2-dimensional mesh generation, in Proc. 4th Symp. Discrete Algorithms, SIAM, Philadelphia, PA, 1993, pp. 83-92.
44. G. STRANG AND G. J. FIX, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
45. S.-H. TENG, Points, Spheres, and Separators: A Unified Geometric Approach to Graph Partitioning, Ph.D. thesis, CMU-CS-91-184, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 1991.
46. P. UNGAR, A theorem on planar graphs, J. London Math. Soc., 26 (1951), pp. 256-262.
47. V. N. VAPNIK AND A. Y. CHERVONENKIS, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl., 16 (1971), pp. 264-280.
48. S. A. VAVASIS, Automatic domain partitioning in three dimensions, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 950-970.
49. S. A. VAVASIS, Stable finite elements for problems with wild coefficients, SIAM J. Numer. Anal., 33 (1996), pp. 890-916.
50. S. A. VAVASIS, QMG Version 1.0, available via ftp://ftp.cs.cornell.edu/pub/vavasis/ qmg1.0.tar.gz and http://www.cs.cornell.edu/home/vavasis/qmg-home.html (1995).