On estimators for eigenvalue/eigenvector approximations

Grubišić, Luka; Ovall, Jeffrey

doi:10.1090/S0025-5718-08-02181-9

On estimators for eigenvalue/eigenvector approximations
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by Luka Grubišić and Jeffrey S. Ovall PDF

Math. Comp. 78 (2009), 739-770 Request permission

Abstract:

We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques—notably, those of gradient recovery type.

References

Gabriel Acosta and Ricardo G. Durán, An optimal Poincaré inequality in $L^1$ for convex domains, Proc. Amer. Math. Soc. 132 (2004), no. 1, 195–202. MR 2021262, DOI 10.1090/S0002-9939-03-07004-7
Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Randolph E. Bank, Hierarchical bases and the finite element method, Acta numerica, 1996, Acta Numer., vol. 5, Cambridge Univ. Press, Cambridge, 1996, pp. 1–43. MR 1624587, DOI 10.1017/S0962492900002610
R. E. Bank. PLTMG: A software package for solving elliptic partial differential equations, users’ guide 9.0. Technical report, University of California, San Diego, 2004.
Randolph E. Bank and Jinchao Xu, Asymptotically exact a posteriori error estimators. I. Grids with superconvergence, SIAM J. Numer. Anal. 41 (2003), no. 6, 2294–2312. MR 2034616, DOI 10.1137/S003614290139874X
Randolph E. Bank and Jinchao Xu, Asymptotically exact a posteriori error estimators. II. General unstructured grids, SIAM J. Numer. Anal. 41 (2003), no. 6, 2313–2332. MR 2034617, DOI 10.1137/S0036142901398751
Christopher Beattie, Galerkin eigenvector approximations, Math. Comp. 69 (2000), no. 232, 1409–1434. MR 1681128, DOI 10.1090/S0025-5718-00-01181-9
M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwendungen 22 (2003), no. 4, 751–756. MR 2036927, DOI 10.4171/ZAA/1170
Carsten Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187–1202. MR 1736895, DOI 10.1051/m2an:1999140
C. Carstensen and S. A. Funken, Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods, East-West J. Numer. Math. 8 (2000), no. 3, 153–175. MR 1807259
Seng-Kee Chua and Richard L. Wheeden, Estimates of best constants for weighted Poincaré inequalities on convex domains, Proc. London Math. Soc. (3) 93 (2006), no. 1, 197–226. MR 2235947, DOI 10.1017/S0024611506015826
Willy Dörfler and Ricardo H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math. 91 (2002), no. 1, 1–12. MR 1896084, DOI 10.1007/s002110100321
Z. Drmač and K. Veselić. New fast and accurate Jacobi SVD algorithm: II. SIAM J. Matrix Anal. Appl., to appear. Preprint LAPACK Working Note 170.
Ricardo G. Durán, Claudio Padra, and Rodolfo Rodríguez, A posteriori error estimates for the finite element approximation of eigenvalue problems, Math. Models Methods Appl. Sci. 13 (2003), no. 8, 1219–1229. MR 1998821, DOI 10.1142/S0218202503002878
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
L. Grubišić. Ritz value estimates and applications in Mathematical Physics. Ph.D. Thesis, Fernuniversität in Hagen, 2005. Available through dissertation.de Verlag im Internet.
Luka Grubišić, On eigenvalue and eigenvector estimates for nonnegative definite operators, SIAM J. Matrix Anal. Appl. 28 (2006), no. 4, 1097–1125. MR 2276556, DOI 10.1137/050626533
L. Grubišić. A posteriori estimates for eigenvalue/vector approximations. PAMM Proc. Appl. Math. Mech., 6(1):59–62, 2006.
L. Grubišić. On Temple–Kato like inequalities and applications. submitted. 2005–Preprint available from http://arxiv.org/abs/math/0511408.
Luka Grubišić and Krešimir Veselić, On weakly formulated Sylvester equations and applications, Integral Equations Operator Theory 58 (2007), no. 2, 175–204. MR 2324886, DOI 10.1007/s00020-007-1482-4
W. Hackbusch, On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method, SIAM J. Numer. Anal. 16 (1979), no. 2, 201–215. MR 526484, DOI 10.1137/0716015
Vincent Heuveline and Rolf Rannacher, A posteriori error control for finite approximations of elliptic eigenvalue problems, Adv. Comput. Math. 15 (2001), no. 1-4, 107–138 (2002). A posteriori error estimation and adaptive computational methods. MR 1887731, DOI 10.1023/A:1014291224961
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Mats G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems, SIAM J. Numer. Anal. 38 (2000), no. 2, 608–625. MR 1770064, DOI 10.1137/S0036142997320164
J.-F. Maitre and F. Musy, The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems, Multigrid methods (Cologne, 1981) Lecture Notes in Math., vol. 960, Springer, Berlin-New York, 1982, pp. 535–544. MR 685787
Dong Mao, Lihua Shen, and Aihui Zhou, Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates, Adv. Comput. Math. 25 (2006), no. 1-3, 135–160. MR 2231699, DOI 10.1007/s10444-004-7617-0
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
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Local problems on stars: a posteriori error estimators, convergence, and performance, Math. Comp. 72 (2003), no. 243, 1067–1097. MR 1972728, DOI 10.1090/S0025-5718-02-01463-1
Klaus Neymeyr, A posteriori error estimation for elliptic eigenproblems, Numer. Linear Algebra Appl. 9 (2002), no. 4, 263–279. MR 1909253, DOI 10.1002/nla.272
Jeffrey S. Ovall, Function, gradient, and Hessian recovery using quadratic edge-bump functions, SIAM J. Numer. Anal. 45 (2007), no. 3, 1064–1080. MR 2318802, DOI 10.1137/060648908
L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
María-Cecilia Rivara, New longest-edge algorithms for the refinement and/or improvement of unstructured triangulations, Internat. J. Numer. Methods Engrg. 40 (1997), no. 18, 3313–3324. MR 1471613, DOI 10.1002/(SICI)1097-0207(19970930)40:18<3313::AID-NME214>3.3.CO;2-R
Lloyd N. Trefethen and Timo Betcke, Computed eigenmodes of planar regions, Recent advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Amer. Math. Soc., Providence, RI, 2006, pp. 297–314. MR 2259116, DOI 10.1090/conm/412/07783
R. Verfürth. A review of a posteriori error estimation and adaptive mesh refinement techniques. Wiley-Teubner Series Advances in Numerical Mathematics. John Wiley & Sons Ltd., Chichester, 1996.
Rüdiger Verfürth, Error estimates for some quasi-interpolation operators, M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 695–713 (English, with English and French summaries). MR 1726480, DOI 10.1051/m2an:1999158