Bounds for eigenvalues and condition numbers in the 𝑝-version of the finite element method

Hu, Ning; Guo, Xian-Zhong; Katz, I.

doi:10.1090/S0025-5718-98-00983-1

Bounds for eigenvalues and condition numbers in the $p$-version of the finite element method
HTML articles powered by AMS MathViewer

by Ning Hu, Xian-Zhong Guo and I. Norman Katz PDF

Math. Comp. 67 (1998), 1423-1450 Request permission

Abstract:

In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the $p$-version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the $p$-version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the $p$-version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like $p^{4(d-1)}$, where $d$ is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that “regardless of the choice of basis, the condition numbers grow like $p^{4d}$ or faster". Numerical results are also presented which verify that our theoretical bounds are correct.

References

I. Babuška, B. A. Szabo, and I. N. Katz, The $p$-version of the finite element method, SIAM J. Numer. Anal. 18 (1981), no. 3, 515–545. MR 615529, DOI 10.1137/0718033
I. Babuška and Manil Suri, The optimal convergence rate of the $p$-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750–776. MR 899702, DOI 10.1137/0724049
Ivo Babuška and Manil Suri, The $p$- and $h$-$p$ versions of the finite element method, an overview, Comput. Methods Appl. Mech. Engrg. 80 (1990), no. 1-3, 5–26. Spectral and high order methods for partial differential equations (Como, 1989). MR 1067939, DOI 10.1016/0045-7825(90)90011-A
I. Babuška, B. Q. Guo, and J. E. Osborn, Regularity and numerical solution of eigenvalue problems with piecewise analytic data, SIAM J. Numer. Anal. 26 (1989), no. 6, 1534–1560. MR 1025104, DOI 10.1137/0726090
I. Babuška, B. Q. Guo, and E. P. Stephan, The $h$-$p$ version of the boundary element method with geometric mesh on polygonal domains, Comput. Methods Appl. Mech. Engrg. 80 (1990), no. 1-3, 319–325. Spectral and high order methods for partial differential equations (Como, 1989). MR 1067959, DOI 10.1016/0045-7825(90)90036-L
I. Babuška, A. Craig, J. Mandel, and J. Pitkäranta, Efficient preconditioning for the $p$-version finite element method in two dimensions, SIAM J. Numer. Anal. 28 (1991), no. 3, 624–661. MR 1098410, DOI 10.1137/0728034
I. Babuška, M. Griebel, and J. Pitkäranta, The problem of selecting the shape functions for a $p$-type finite element, Internat. J. Numer. Methods Engrg. 28 (1989), no. 8, 1891–1908. MR 1008139, DOI 10.1002/nme.1620280813
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
Randolph E. Bank and Todd Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35–51. MR 595040, DOI 10.1090/S0025-5718-1981-0595040-2
D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the $V$-cycle, SIAM J. Numer. Anal. 20 (1983), no. 5, 967–975. MR 714691, DOI 10.1137/0720066
James H. Bramble and Joseph E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311–329. MR 906174, DOI 10.1090/S0025-5718-1987-0906174-X
N. Hu, Multi-p processes: Iterative algorithms and preconditionings for the p-version of finite element analysis, Doctoral dissertation, Department of Systems Science and Mathematics, Washington University inSt. Louis, August, 1994.
N. Hu, X. Guo and I.N. Katz, Lower and upper bounds for eigenvalues and condition numbers in the p-version of the finite element method, SIAM annual meeting, 1995, Charlotte, North Carolina.
Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
Claes Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1987. MR 925005
Jean-François Maitre and Olivier Pourquier, Conditionnements et préconditionnements diagonaux pour la $p$-version des méthodes d’éléments finis pour des problèmes elliptiques du second ordre, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 6, 583–586 (French, with English and French summaries). MR 1270086
Jean-François Maitre and Olivier Pourquier, Condition number and diagonal preconditioning: comparison of the $p$-version and the spectral element methods, Numer. Math. 74 (1996), no. 1, 69–84. MR 1400216, DOI 10.1007/s002110050208
J. T. Marti, Introduction to Sobolev spaces and finite element solution of elliptic boundary value problems, Computational Mathematics and Applications, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. MR 918971
G. Sansone, Orthogonal functions, Pure and Applied Mathematics, Vol. IX, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959. Revised English ed; Translated from the Italian by A. H. Diamond; with a foreword by E. Hille. MR 0103368
Barna Szabó and Ivo Babuška, Finite element analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. MR 1164869
Stress Check, http://www.esrd.com, Engineering Software Research and Development, Inc., St. Louis, MO 63117, 1994.
Elwood T. Olsen and Jim Douglas Jr., Bounds on spectral condition numbers of matrices arising in the $p$-version of the finite element method, Numer. Math. 69 (1995), no. 3, 333–352. MR 1312785, DOI 10.1007/s002110050096