Superconvergence postprocessing for eigenvalues

Computational Methods in Applied Mathematics

Volumn 2, Issue 2, 2002, Pages 171-185

  Racheva, Milena R ^a   Andreev, Andrey B ^a  

^a TECHNICAL UNIVERSITY OF GABROVO   (Bulgaria)

Author keywords

Eigenvalue problems; Finite element method; Postprocessing; Superconvergence

Indexed keywords

EID: 85008503246     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2002-0011     Document Type: Article

References (29)

1. A. B. Andreev, Superconvergence of the gradient of finite element eigenfunctions, C. R. Acad. Bulgare Sci., 43 (1990), pp. 9-11.
2. A. B. Andreev and R. D. Lazarov, Superconvergence of the gradient for quadratic triangular finite elements, Numer. Methods for PDEs, 4 (1988), pp. 15-32.
3. I. Babuska and A. K. Aziz, Survey lectures on the mathematics foundations of the finite element method, in: The Mathematical Foundations of the Finite Element Method (A. K. Aziz, ed.), Academic Press, 1972, pp. 1-359.
4. I. Babuska and J. Osborn, Eigenvalue Problems, Handbook of Numer. Anal., Vol.II, North-Holland, Amsterdam, 1991.
5. I. Babuska, T. Strouboulis, C. S. Upadhyay, and S. K. Gangaraj, Validation of recipes for the recovery of stresses and derivatives by a computer-based approach, Math. Comput. Mode., 20 (1994), p. 45.
6. I. Babuska, T. Strouboulis, C. S. Upadhyay, and S. K. Gangaraj, Computer-based proof of the existence of superconvergence points in, the finite element method; Superconvergence of the derivatives in finite element solutions of Laplace's, Poisson's and the elasticity equations, Numer. Methods for PDEs, 12 (1996), pp. 347-392.
7. B. P. Belinskiy and J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Mathematics, LVI (Sept. 1998), No. 3, pp. 521-541.
8. S. C. Brener and L. R. Scott, The Mathematical Theory of Finite Elelment Methods, Texts in Appl. Math., Springer-Verlag, 1994.
9. F. Chatelin, The spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators, SIAM Rev., 23 (1981), pp. 495-522.
10. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.
11. K. Eriksson, D. Ester, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
12. B. Garcia Archilla, J. Novo, and E. S. Titi, Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds, SIAM J. Numer. Anal., 35 (1998), pp. 941-972.
13. B. Garcia Archilla, J. Novo, and E. S. Titi, An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations, Math. of Comput., 68 (1999), No. 227, pp. 893-911.
14. B. Garcia Archilla and E. S. Titi, Postprocessing the Galerkin Method: The finite element case, SIAM J. Numer. Anal., 37 (2000), pp. 470-499.
15. G. Goodsell and J. R. Whiteman, A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations, Internat. J. Numer. Methods. Eng., 27 (1989), pp. 469-481.
16. G. H. Handelman and J. B. Keller, Small vibrations of a slightly stiff pendulum, in: Proc. Fourth U. S. Natl. Congress of Appl. Mech., 1962, pp. 195-202.
17. M. Krizek and P. Neittaanmaki, Superconvergence phenomenon in the finite element method arising from averaging gradients, Numer. Math., 45 (1984), pp. 105-116.
18. C. R. Laing, A. McRobie, and J. M. T. Thompson, The post-processed Galerkin method applied to nonlinear shell vibrations, Dynam. Stability Systems., 14 (1999), pp. 166-181.
19. M. G. Larson, A posteriori and a priori error analysis for finite element approximations of self-adjoint eigenvalue problems, SIAM J. Numer. Anal., 38 (2000), pp. 608-625.
20. M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal., 23 (1986), No. 3, pp. 562-580.
21. J. Pierce and R. S. Varga, Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: Improved error bounds for eigenfunctions, Numer. Math., 19 (1972), pp. 155-169.
22. K. Rektorys, Variational Methods in Mathematics, Science and Engineering, SNTL - Publishers in Technical Literature, Prague, 1980.
23. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, New York, 1993.
24. A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal., 33 (1996), pp. 505-521.
25. G. Strang and G. J. Fix, An Analysis of the Finite Elemente Method, Prentice - Hall, Englewood Cliffs, NJ, 1973.
26. R. Verfurth, A posteriori error estimates for nonlinear problems. Finite elements discretizations of elliptic equations, Math. Comp., 62 (1994), pp. 445-475.
27. J. Xu, Two-mesh discretization techniques for linear and nonlinear problems, SIAM J. Numer. Anal., 33 (1966), pp. 1759-1777.
28. J. Xu and A. Zhou, A two-mesh discretization scheme for eigenvalue problems, To appear in Math. of Computation.
29. O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch-recovery (SPR) and adaptive finite element refinement, Comput. Methods Appl. Mech. Eng., 101 (1992), pp. 207-224.