Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators

Andrey Andreev; Milena Racheva

Applications of Mathematics (2014)

  • Volume: 59, Issue: 4, page 371-390
  • ISSN: 0862-7940

Abstract

top
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 λ l , 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.

How to cite

top

Andreev, Andrey, and Racheva, Milena. "Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators." Applications of Mathematics 59.4 (2014): 371-390. <http://eudml.org/doc/261879>.

@article{Andreev2014,
abstract = {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 $\lambda _l$, $l = 2,3,\ldots $, 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.},
author = {Andreev, Andrey, Racheva, Milena},
journal = {Applications of Mathematics},
keywords = {eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; two-sided bounds; fourth-order problems; numerical experiments; plate bending},
language = {eng},
number = {4},
pages = {371-390},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators},
url = {http://eudml.org/doc/261879},
volume = {59},
year = {2014},
}

TY - JOUR
AU - Andreev, Andrey
AU - Racheva, Milena
TI - Two-sided bounds of eigenvalues of second- and fourth-order elliptic operators
JO - Applications of Mathematics
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 4
SP - 371
EP - 390
AB - 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 $\lambda _l$, $l = 2,3,\ldots $, 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.
LA - eng
KW - eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; eigenvalue problem; nonconforming finite element method; conforming finite element method; postprocessing; lower bound; two-sided bounds; fourth-order problems; numerical experiments; plate bending
UR - http://eudml.org/doc/261879
ER -

References

top
  1. Adini, A., Clough, R., Analysis of Plate Bending by the Finite Element Method, NSF Report G. 7337 (1961). (1961) 
  2. Andreev, A. B., Lazarov, R. D., Racheva, M. R., 10.1016/j.cam.2004.12.015, J. Comput. Appl. Math. 182 (2005), 333-349. (2005) MR2147872DOI10.1016/j.cam.2004.12.015
  3. Andreev, A. B., Racheva, M. R., Superconvergent FE postprocessing for eigenfunctions, C. R. Acad. Bulg. Sci. 55 (2002), 17-22. (2002) Zbl1007.65087MR1885694
  4. Andreev, A. B., Racheva, M. R., Lower bounds for eigenvalues by nonconforming FEM on convex domain, Application of Mathematics in Technical and Natural Sciences Proceedings of the 2nd international conference, Sozopol, Bulgaria, 2010. AIP Conf. Proc. 1301 Amer. Inst. Phys., Melville (2010), 361-369 M. Todorov et al. (2010), 361-369. (2010) Zbl1232.35105MR2810107
  5. Andreev, A. B., Racheva, M. R., Properties and estimates of an integral type nonconforming finite element, Large-Scale Scientific Computing 8th international conference, LSSC 2011, Sozopol, Bulgaria, 2011. Lecture Notes in Computer Science 7116, 2012, pp. 252-532 Springer, Berlin I. Lirkov et al. MR2955161
  6. Andreev, A. B., Racheva, M. R., Lower bounds for eigenvalues and postprocessing by an integral type nonconforming FEM, Sib. Zh. Vychisl. Mat. 15 235-249 (2012), Russian Numer. Analysis Appl. 5 (2012), 191-203. (2012) Zbl1299.35217
  7. Andreev, A. B., Racheva, M. R., Tsanev, G. S., A Nonconforming Finite Element with Integral Type Bubble Function, Proceedings of 5th Annual Meeting of the BG. Section of SIAM'10 (2010), 3-6. (2010) 
  8. Armentano, M. G., Durán, R. G., Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, ETNA, Electron. Trans. Numer. Anal. (electronic only) 17 (2004), 93-101. (2004) Zbl1065.65127MR2040799
  9. Babuška, I., Kellogg, R. B., Pitkäranta, J., 10.1007/BF01399326, Numer. Math. 33 (1979), 447-471. (1979) Zbl0423.65057MR0553353DOI10.1007/BF01399326
  10. Babuška, I., Osborn, J., Eigenvalue problems, Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part 1) J.-L. Lions, P. G. Ciarlet North-Holland, Amsterdam (1991), 641-787. (1991) MR1115240
  11. Brenner, S. C., Scott, R. L., 10.1007/978-1-4757-4338-8_7, Texts in Applied Mathematics 15 Springer, New York (1994). (1994) Zbl0804.65101MR1278258DOI10.1007/978-1-4757-4338-8_7
  12. Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications. Vol. 4 North-Holland, Amsterdam (1978). (1978) Zbl0383.65058MR0520174
  13. Ciarlet, P. G., Basic error estimates for elliptic problems, Handbook of Numerical Analysis. Vol. II: Finite Element Methods (Part 1) P. G. Ciarlet et al. North-Holland, Amsterdam (1991). (1991) Zbl0875.65086MR1115237
  14. Crouzeix, M., Raviart, P. A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Franc. Automat. Inform. Rech. Operat. 7 (1973), 33-75. (1973) MR0343661
  15. Forsythe, G. E., 10.2140/pjm.1955.5.691, Pac. J. Math. 5 (1955), 691-702. (1955) Zbl0068.10304MR0073048DOI10.2140/pjm.1955.5.691
  16. Grisvard, P., Singularities in Boundary Problems, MASSON and Springer, Berlin (1985). (1985) 
  17. Huang, H. T., Li, Z. C., Lin, Q., 10.1016/j.cam.2007.06.011, J. Comput. Appl. Math. 217 (2008), 9-27. (2008) Zbl1147.65090MR2427427DOI10.1016/j.cam.2007.06.011
  18. Lascaux, P., Lesaint, P., Some nonconforming finite elements for the plate bending problem, Rev. Franc. Automat. Inform. Rech. Operat. , Analyse numer. R-1 9-53 (1975). (1975) Zbl0319.73042MR0423968
  19. Lin, Q., Lin, J. F., Finite Element Methods: Accuracy and Improvement, Science Press, Beijing (2006). (2006) 
  20. Lin, Q., Huang, H. T., Li, Z. C., 10.1090/S0025-5718-08-02098-X, Math. Comput. 77 (2008), 2061-2084. (2008) Zbl1198.65228MR2429874DOI10.1090/S0025-5718-08-02098-X
  21. Lin, Q., Tobiska, L., Zhou, A., 10.1093/imanum/drh008, IMA J. Numer. Anal. 25 (2005), 160-181. (2005) Zbl1068.65122MR2110239DOI10.1093/imanum/drh008
  22. Lin, Q., Xie, H., The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods, Math. Pract. Theory 42 (2012), 219-226 Chinese. (2012) Zbl1289.65251MR3013284
  23. Lin, Q., Xie, H., Luo, F., Li, Y., Yang, Y., Stokes eigenvalue approximations from below with nonconforming mixed finite element methods, Math. Pract. Theory 40 (2010), 157-168. (2010) MR2768711
  24. Lin, Q., Xie, H., Xu, J., Lower bounds of the discretization for piecewise polynomials, http://arxiv.org/abs/1106.4395 (2011). (2011) MR3120579
  25. Liu, H. P., Yan, N. N., Four finite element solutions and comparison of problem for the Poisson equation eigenvalue, Chin. J. Numer. Math. Appl. 27 27-39 (2005), 81-91. (2005) Zbl1106.65327MR2159418
  26. Luo, F., Lin, Q., Xie, H., 10.1007/s11425-012-4382-2, Sci. China, Math. 55 (2012), 1069-1082. (2012) Zbl1261.65112MR2912496DOI10.1007/s11425-012-4382-2
  27. Morley, L. S. D., 10.1017/S0001925900004546, Aero. Quart. 19 (1968), 149-169. (1968) DOI10.1017/S0001925900004546
  28. Nicaise, S., 10.1137/S0036142903437787, SIAM J. Numer. Anal. (electronic) 43 (2005), 1481-1503. (2005) Zbl1103.65110MR2182137DOI10.1137/S0036142903437787
  29. Racheva, M. R., Andreev, A. B., 10.2478/cmam-2002-0011, Comput. Methods Appl. Math. 2 (2002), 171-185. (2002) Zbl1012.65113MR1930846DOI10.2478/cmam-2002-0011
  30. Rannacher, R., 10.1007/BF01396493, Numer. Math. 33 (1979), 23-42. (1979) Zbl0394.65035MR0545740DOI10.1007/BF01396493
  31. Rannacher, R., Turek, S., 10.1002/num.1690080202, Numer. Methods Partial Differ. Equations 8 (1992), 97-111. (1992) Zbl0742.76051MR1148797DOI10.1002/num.1690080202
  32. Shi, Z.-C., On the error estimates of Morley element, Math. Numer. Sin. Chinese 12 (1990), 113-118 translation in Chinese J. Numer. Math. Appl. 12 (1990), 102-108. (1990) Zbl0850.73337MR1070298
  33. Strang, G., Fix, G. J., An Analysis of the Finite Element Method, Prentice-Hall Series in Automatic Computation Prentice-Hall, Englewood Cliffs (1973). (1973) Zbl0356.65096MR0443377
  34. Wang, M., Shi, Z.-C., Xu, J., Some n -rectangle nonconforming elements for fourth order elliptic equations, J. Comput. Math. 25 (2007), 408-420. (2007) Zbl1142.65451MR2337403
  35. Wang, L., Wu, Y., Xie, X., 10.1002/num.21723, Numer. Methods Partial Differ. Equations 29 (2013), 721-737. (2013) MR3039784DOI10.1002/num.21723
  36. Xu, J., Zhou, A., 10.1090/S0025-5718-99-01180-1, Math. Comput. 70 (2001), 17-25. (2001) Zbl0959.65119MR1677419DOI10.1090/S0025-5718-99-01180-1
  37. Yang, Y. D., A posteriori error estimates in Adini finite element for eigenvalue problems, J. Comput. Math. 18 (2000), 413-418. (2000) Zbl0957.65092MR1773912
  38. Yang, Y. D., Zhang, Z. M., Lin, F. B., 10.1007/s11425-009-0198-0, Sci. China, Math. 53 (2010), 137-150. (2010) Zbl1187.65125MR2594754DOI10.1007/s11425-009-0198-0
  39. Zhang, H. Q., Wang, M., The Mathematical Theory of Finite Elements, Science Press, Beijing (1991). (1991) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.