zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A two-scale discretization scheme for mixed variational formulation of eigenvalue problems. (English) Zbl 1246.65220
Summary: We discuss highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid $K^h$ is reduced to the solution of an eigenvalue problem on a much coarser grid $K^H$ and the solution of a linear algebraic system on the fine grid $K^h$. Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking $H = O(\root 4 \of {h})$, and when using the $P_{k+1} - P_k$ element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking $H = O(\root 3 \of {h})$. Finally, numerical experiments are presented to support the theoretical analysis.

MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
WorldCat.org
Full Text: DOI
References:
[1] J. Xu, “A new class of iterative methods for nonselfadjoint or indefinite problems,” SIAM Journal on Numerical Analysis, vol. 29, no. 2, pp. 303-319, 1992. · Zbl 0756.65050 · doi:10.1137/0729020
[2] J. Xu, “A novel two-grid method for semilinear elliptic equations,” SIAM Journal on Scientific Computing, vol. 15, no. 1, pp. 231-237, 1994. · Zbl 0795.65077 · doi:10.1137/0915016
[3] J. Xu, “Two-grid discretization techniques for linear and nonlinear PDEs,” SIAM Journal on Numerical Analysis, vol. 33, no. 5, pp. 1759-1777, 1996. · Zbl 0860.65119 · doi:10.1137/S0036142992232949
[4] M. Cai, M. Mu, and J. Xu, “Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach,” SIAM Journal on Numerical Analysis, vol. 47, no. 5, pp. 3325-3338, 2009. · Zbl 1213.76131 · doi:10.1137/080721868
[5] Y. He, J. Xu, A. Zhou, and J. Li, “Local and parallel finite element algorithms for the Stokes problem,” Numerische Mathematik, vol. 109, no. 3, pp. 415-434, 2008. · Zbl 1145.65097 · doi:10.1007/s00211-008-0141-2
[6] J. Li, “Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1470-1481, 2006. · Zbl 1151.76528 · doi:10.1016/j.amc.2006.05.034
[7] M. Mu and J. Xu, “A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow,” SIAM Journal on Numerical Analysis, vol. 45, no. 5, pp. 1801-1813, 2007. · Zbl 1146.76031 · doi:10.1137/050637820
[8] C.-S. Chien and B.-W. Jeng, “A two-grid discretization scheme for semilinear elliptic eigenvalue problems,” SIAM Journal on Scientific Computing, vol. 27, no. 4, pp. 1287-1340, 2006. · Zbl 1095.65100 · doi:10.1137/030602447
[9] J. Xu and A. Zhou, “A two-grid discretization scheme for eigenvalue problems,” Mathematics of Computation, vol. 70, no. 233, pp. 17-25, 2001. · Zbl 0959.65119 · doi:10.1090/S0025-5718-99-01180-1
[10] Q. Lin and G. Q. Xie, “Acceleration of FEA for eigenvalue problems,” Bulletin of Science, vol. 26, pp. 449-452, 1981.
[11] I. H. Sloan, “Iterated Galerkin method for eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 13, no. 5, pp. 753-760, 1976. · Zbl 0359.65052 · doi:10.1137/0713061
[12] X. Dai and A. Zhou, “Three-scale finite element discretizations for quantum eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 46, no. 1, pp. 295-324, 2007/08. · Zbl 1160.65060 · doi:10.1137/06067780X
[13] X. Gong, L. Shen, D. Zhang, and A. Zhou, “Finite element approximations for Schrödinger equations with applications to electronic structure computations,” Journal of Computational Mathematics, vol. 26, no. 3, pp. 310-323, 2008. · Zbl 1174.65047
[14] H. Chen, F. Liu, and A. Zhou, “A two-scale higher-order finite element discretization for Schrödinger equation,” Journal of Computational Mathematics, vol. 27, no. 2-3, pp. 315-337, 2009. · Zbl 1212.65432 · doi:10.4208/jcm.2009.27.4.018
[15] K. Kolman, “A two-level method for nonsymmetric eigenvalue problems,” Acta Mathematicae Applicatae Sinica, vol. 21, no. 1, pp. 1-12, 2005. · Zbl 1084.65109 · doi:10.1007/s10255-005-0209-z
[16] Y. Yang and X. Fan, “Generalized Rayleigh quotient and finite element two-grid discretization schemes,” Science in China A, vol. 52, no. 9, pp. 1955-1972, 2009. · Zbl 1188.65151 · doi:10.1007/s11425-009-0016-8
[17] H. Bi and Y. Yang, “A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9669-9678, 2011. · Zbl 1222.65121 · doi:10.1016/j.amc.2011.04.051
[18] Q. Li and Y. Yang, “A two-grid discretization scheme for the Steklov eigenvalue problem,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 129-139, 2011. · Zbl 1220.65160 · doi:10.1007/s12190-010-0392-9
[19] H. Chen, S. Jia, and H. Xie, “Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems,” Applications of Mathematics, vol. 54, no. 3, pp. 237-250, 2009. · Zbl 1212.65431 · doi:10.1007/s10492-009-0015-7 · eudml:37818
[20] Y. D. Yang, “Iterated Galerkin method and Rayleigh quotient for accelerating convergence of eigenvalue problems,” Chinese Journal of Engineering Mathematics, vol. 25, no. 3, pp. 480-488, 2008. · Zbl 1174.65517
[21] Y. Yang and H. Bi, “Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 49, no. 4, pp. 1602-1624, 2011. · Zbl 1236.65143 · doi:10.1137/100810241
[22] L. N. Trefethen and D. Bau,, Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1997. · Zbl 0874.65013
[23] I. Babu\vska and J. Osborn, “Eigenvalue problems,” in Finite Element Methods(Part 1), Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., pp. 641-787, Elsevier Science, North Holland, The Netherlands, 1991.
[24] D. Boffi, F. Brezzi, and L. Gastaldi, “On the convergence of eigenvalues for mixed formulations,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, vol. 25, no. 1-2, pp. 131-154, 1997. · Zbl 1003.65052 · numdam:ASNSP_1997_4_25_1-2_131_0 · eudml:84281
[25] B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, “Eigenvalue approximation by mixed and hybrid methods,” Mathematics of Computation, vol. 36, no. 154, pp. 427-453, 1981. · Zbl 0472.65080 · doi:10.2307/2007651
[26] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Theory and Algorithms, Springer, Heidelberg, Germany, 1986. · Zbl 0585.65077
[27] C. Bernardi and G. Raugel, “Analysis of some finite elements for the Stokes problem,” Mathematics of Computation, vol. 44, no. 169, pp. 71-79, 1985. · Zbl 0563.65075 · doi:10.2307/2007793
[28] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15, Springer, New York, NY, USA, 1991. · Zbl 0788.73002
[29] R. Stenberg, “Analysis of mixed finite elements methods for the Stokes problem: a unified approach,” Mathematics of Computation, vol. 42, no. 165, pp. 9-23, 1984. · Zbl 0535.76037 · doi:10.2307/2007557
[30] R. B. Kellogg and J. E. Osborn, “A regularity result for the Stokes problem in a convex polygon,” Journal of Functional Analysis, vol. 21, no. 4, pp. 397-431, 1976. · Zbl 0317.35037 · doi:10.1016/0022-1236(76)90035-5
[31] D. N. Arnold, F. Brezzi, and M. Fortin, “A stable finite element for the Stokes equations,” Calcolo, vol. 21, no. 4, pp. 337-344, 1984. · Zbl 0593.76039 · doi:10.1007/BF02576171
[32] P. G. Ciarlet, “Basic error estimates for elliptic problems,” in Finite Element Methods(Part1), Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., vol. 3, pp. 21-343, Elsevier Science, North Holland, The Netherlands, 1991.
[33] R. Verfürth, “Error estimates for a mixed finite element approximation of the Stokes equations,” RAIRO Analyse Numérique, vol. 18, no. 2, pp. 175-182, 1984. · Zbl 0557.76037 · eudml:193431
[34] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, “Vector potentials in three-dimensional non-smooth domains,” Mathematical Methods in the Applied Sciences, vol. 21, no. 9, pp. 823-864, 1998. · Zbl 0914.35094 · doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
[35] M. Costabel, “A coercive bilinear form for Maxwell’s equations,” Journal of Mathematical Analysis and Applications, vol. 157, no. 2, pp. 527-541, 1991. · Zbl 0738.35095 · doi:10.1016/0022-247X(91)90104-8
[36] P. Ciarlet, Jr., “Augmented formulations for solving Maxwell equations,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 2-5, pp. 559-586, 2005. · Zbl 1063.78018 · doi:10.1016/j.cma.2004.05.021
[37] M. Costabel and M. Dauge, “Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements,” Numerische Mathematik, vol. 93, no. 2, pp. 239-277, 2002. · Zbl 1019.78009 · doi:10.1007/s002110100388
[38] A. Buffa, P. Ciarlet, Jr., and E. Jamelot, “Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements,” Numerische Mathematik, vol. 113, no. 4, pp. 497-518, 2009. · Zbl 1180.78048 · doi:10.1007/s00211-009-0246-2
[39] F. Kikuchi, “Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,” Computer Methods in Applied Mechanics and Engineering, vol. 64, pp. 509-521, 1987. · Zbl 0644.65087 · doi:10.1016/0045-7825(87)90053-3
[40] P. Ciarlet, Jr. and V. Girault, “inf-sup condition for the 3D, P2-iso-P1 Taylor-Hood finite element application to Maxwell equations,” Comptes Rendus Mathématique, vol. 335, no. 10, pp. 827-832, 2002. · Zbl 1021.78009 · doi:10.1016/S1631-073X(02)02564-5
[41] P. Ciarlet, Jr. and G. Hechme, “Computing electromagnetic eigenmodes with continuous Galerkin approximations,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 2, pp. 358-365, 2008. · Zbl 1194.78053 · doi:10.1016/j.cma.2008.08.005
[42] D. Boffi, P. Fernandes, L. Gastaldi, and I. Perugia, “Computational models of electromagnetic resonators: analysis of edge element approximation,” SIAM Journal on Numerical Analysis, vol. 36, no. 4, pp. 1264-1290, 1999. · Zbl 1025.78014 · doi:10.1137/S003614299731853X