Two-sided bounds of eigenvalues of second-and fourth-order elliptic operators
Tóm tắt
This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues λl, l = 2, 3, …, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.
Tài liệu tham khảo
A. Adini, R. Clough: Analysis of Plate Bending by the Finite Element Method. NSF Report G. 7337, 1961.
A. B. Andreev, R. D. Lazarov, M. R. Racheva: Postprocessing and higher order convergence of mixed finite element approximations of biharmonic eigenvalue problems. J. Comput. Appl. Math. 182 (2005), 333–349.
A. B. Andreev, M. R. Racheva: Superconvergent FE postprocessing for eigenfunctions. C. R. Acad. Bulg. Sci. 55 (2002), 17–22.
A. B. Andreev, M. R. Racheva: Lower bounds for eigenvalues by nonconforming FEM on convex domain. Application of Mathematics in Technical and Natural Sciences (M. Todorov et al., ed.). Proceedings of the 2nd international conference, Sozopol, Bulgaria, 2010. AIP Conf. Proc. 1301, Amer. Inst. Phys., Melville, 2010, pp. 361–369.
A. B. Andreev, M. R. Racheva: Properties and estimates of an integral type nonconforming finite element. Large-Scale Scientific Computing (I. Lirkov et al., ed.). 8th international conference, LSSC 2011, Sozopol, Bulgaria, 2011. Lecture Notes in Computer Science 7116, 2012, pp. 252–532, Springer, Berlin.
A. B. Andreev, M. R. Racheva: Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM. Sib. Zh. Vychisl. Mat. 15 (2012), 235–249 (In Russian.); Numer. Analysis Appl. 5 (2012), 191–203.
A. B. Andreev, M. R. Racheva, G. S. Tsanev: A Nonconforming Finite Element with Integral Type Bubble Function. Proceedings of 5th Annual Meeting of the BG. Section of SIAM’10, 2010, pp. 3–6.
M. G. Armentano, R. G. Durán: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. ETNA, Electron. Trans. Numer. Anal. (electronic only) 17 (2004), 93–101.
I. Babuška, R. B. Kellogg, J. Pitkäranta: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33 (1979), 447–471.
I. Babuška, J. Osborn: Eigenvalue problems. Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part 1) (J. -L. Lions, P. G. Ciarlet, ed.). North-Holland, Amsterdam, 1991, pp. 641–787.
S. C. Brenner, R. L. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15, Springer, New York, 1994.
P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications. Vol. 4, North-Holland, Amsterdam, 1978.
P. G. Ciarlet: Basic error estimates for elliptic problems. Handbook of Numerical Analysis. Vol. II: Finite Element Methods (Part 1) (P. G. Ciarlet et al., eds.). North-Holland, Amsterdam, 1991.
M. Crouzeix, P. A. Raviart: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33–75.
G. E. Forsythe: Asymptotic lower bounds for the fundamental frequency of convex membranes. Pac. J. Math. 5 (1955), 691–702.
P. Grisvard: Singularities in Boundary Problems. MASSON and Springer, Berlin, 1985.
H. T. Huang, Z. C. Li, Q. Lin: New expansions of numerical eigenvalues by finite elements. J. Comput. Appl. Math. 217 (2008), 9–27.
P. Lascaux, P. Lesaint: Some nonconforming finite elements for the plate bending problem. Rev. Franc. Automat. Inform. Rech. Operat. 9, Analyse numer. R-1 (1975), 9–53.
Q. Lin, J. F. Lin: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing, 2006.
Q. Lin, H. T. Huang, Z. C. Li: New expansions of numerical eigenvalues for −Δu = λu by nonconforming elements. Math. Comput. 77 (2008), 2061–2084.
Q. Lin, L. Tobiska, A. Zhou: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25 (2005), 160–181.
Q. Lin, H. Xie: The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods. Math. Pract. Theory 42 (2012), 219–226. (In Chinese. )
Q. Lin, H. Xie, F. Luo, Y. Li, Y. Yang: Stokes eigenvalue approximations from below with nonconforming mixed finite element methods. Math. Pract. Theory 40 (2010), 157–168.
Q. Lin, H. Xie, J. Xu: Lower bounds of the discretization for piecewise polynomials. http://arxiv.org/abs/1106.4395 (2011).
H. P. Liu, N. N. Yan: Four finite element solutions and comparison of problem for the Poisson equation eigenvalue. Chin. J. Numer. Math. Appl. 27 (2005), 81–91.
F. Luo, Q. Lin, H. Xie: Computing the lower and upper bounds of Laplace eigenvalue problem by combining conforming and nonconforming finite element methods. Sci. China, Math. 55 (2012), 1069–1082.
L. S. D. Morley: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19 (1968), 149–169.
S. Nicaise: A posteriori error estimations of some cell-centered finite volume methods. SIAM J. Numer. Anal. (electronic) 43 (2005), 1481–1503.
M. R. Racheva, A. B. Andreev: Superconvergence postprocessing for eigenvalues. Comput. Methods Appl. Math. 2 (2002), 171–185.
R. Rannacher: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33 (1979), 23–42.
R. Rannacher, S. Turek: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97–111.
Z. -C. Shi: On the error estimates of Morley element. Math. Numer. Sin. 12 (1990), 113–118 (In Chinese.); translation in Chinese J. Numer. Math. Appl. 12 (1990), 102–108.
G. Strang, G. J. Fix: An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, 1973.
M. Wang, Z. -C. Shi, J. Xu: Some n-rectangle nonconforming elements for fourth order elliptic equations. J. Comput. Math. 25 (2007), 408–420.
L. Wang, Y. Wu, X. Xie: Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems. Numer. Methods Partial Differ. Equations 29 (2013), 721–737.
J. Xu, A. Zhou: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70 (2001), 17–25.
Y. D. Yang: A posteriori error estimates in Adini finite element for eigenvalue problems. J. Comput. Math. 18 (2000), 413–418.
Y. D. Yang, Z. M. Zhang, F. B. Lin: Eigenvalue approximation from below using non-conforming finite elements. Sci. China, Math. 53 (2010), 137–150.
H. Q. Zhang, M. Wang: The Mathematical Theory of Finite Elements. Science Press, Beijing, 1991.