article

Acceleration of Weak Galerkin Methods for the Laplacian Eigenvalue Problem

Authors:

Zhimin ZhangAuthors Info & Claims

Journal of Scientific Computing, Volume 79, Issue 2

Pages 914 - 934

https://doi.org/10.1007/s10915-018-0877-5

Published: 01 May 2019 Publication History

Abstract

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis.

References

[1]

Armentano, M.G., Duran, R.G.: Asymptotic lower bounds for eigenvalues by nonconforming finite element methods. Electron. Trans. Numer. Anal. 17, 93---101 (2004)

[2]

Babuska, I., Osborn, J.: Handbook of Numerical Analysis, Vol II, Part 1. Elsevier, North-Holland (1991)

[3]

Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer. 19, 1---120 (2010)

[4]

Bramble, J.H., Hubbard, B.E.: Effects of boundary regularity on the discretization error in the fixed membrane eigenvalue problem. SIAM J. Numer. Anal. 5(4), 835---863 (1968)

[5]

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15. Springer New York, New York (2008)

[6]

Chatelin, F.: Spectral approximation of linear operators. In: Spectr. Approx. Linear Oper. pp. xxvii+458. Society for Industrial and Applied Mathematics (2011)

[7]

Chen, H., Jia, S., Xie, H.: Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems. Appl. Math. 54(3), 237---250 (2009)

Digital Library

[8]

Chen, H., Jia, S., Xie, H.: Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem. Appl. Numer. Math. 61(4), 615---629 (2011)

Digital Library

[9]

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)

Digital Library

[10]

Feng, X., Weng, Z., Xie, H.: Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem. Appl. Math. 59(6), 615---630 (2014)

[11]

Grebenkov, D.S., Nguyen, B.-T.: Geometrical structure of Laplacian eigenfunctions. SIAM Rev. 55(4), 601---667 (2013)

Digital Library

[12]

Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman Advanced Publishing Progr. Pitman Advanced Publishing Program, Boston (1985)

[13]

Henning, P., Målqvist, A., Peterseim, D.: Two-level discretization techniques for ground state computations of Bose---Einstein condensates. SIAM J. Numer. Anal. 52(4), 1525---1550 (2014)

[14]

Hu, J., Huang, Y., Lin, Q.: Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods. J. Sci. Comput. 61(1), 196---221 (2014)

Digital Library

[15]

Hu, X., Cheng, X.: Acceleration of a two-grid method for eigenvalue problems. Math. Comput. 80(275), 1287---1301 (2011)

[16]

Larson, M.G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38(2), 608---625 (2000)

Digital Library

[17]

Li, Q., Wang, J.: Weak Galerkin finite element methods for parabolic equations. Numer. Methods Partial Differ. Equ. 29(6), 1---21 (2013)

[18]

Liu, X.: A framework of verified eigenvalue bounds for self-adjoint differential operators. Appl. Math. Comput. 267, 341---355 (2015)

Digital Library

[19]

Luo, F., Lin, Q., Xie, H.: Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods. Sci. China Math. 55(5), 1069---1082 (2012)

[20]

Marion, M., Xu, J.: Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal. 32(4), 1170---1184 (1995)

Digital Library

[21]

Mu, L., Wang, J., Ye, X.: A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J. Comput. Phys. 273, 327---342 (2014)

Digital Library

[22]

Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numer. Methods Partial Differ. Equ. 30(3), 1003---1029 (2014)

[23]

Mu, L., Wang, J., Ye, X., Zhang, S.: A C0-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 59(2), 473---495 (2014)

[24]

Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65(1), 363---386 (2015)

Digital Library

[25]

Racheva, M.R., Andreev, A.B.: Superconvergence postprocessing for eigenvalues. Comput. Methods Appl. Math. 2(2), 171---185 (2002)

[26]

Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2017)

[27]

Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241(1), 103---115 (2013)

Digital Library

[28]

Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comput. 83(289), 2101---2126 (2014)

[29]

Weinberger, H.F.: Lower bounds for higher eigenvalues by finite difference methods. Pac. J. Math. 8(2), 339---368 (1958)

[30]

Xie, H., Yin, X.: Acceleration of stabilized finite element discretizations for the Stokes eigenvalue problem. Adv. Comput. Math. 41(4), 799---812 (2015)

Digital Library

[31]

Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15(1), 231---237 (1994)

Digital Library

[32]

Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759---1777 (1996)

Digital Library

[33]

Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70(233), 17---26 (1999)

Digital Library

[34]

Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69(231), 881---909 (2000)

Digital Library

[35]

Yang, Y., Bi, H.: Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 49(4), 1602---1624 (2011)

Digital Library

[36]

Q. Zhai, H. Xie, R. Zhang, Z. Zhang. The weak Galerkin method for elliptic eigenvalue problems. Commun. Comput. Phys. Accepted. arXiv:1508.05304

[37]

Zhang, R., Zhai, Q.: A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J. Sci. Comput. 64(2), 559---585 (2015)

Digital Library

[38]

Zhou, J., Hu, X., Zhong, L., Shu, S., Chen, L.: Two-Grid methods for Maxwell eigenvalue problems. SIAM J. Numer. Anal. 52(4), 2027---2047 (2014)

Cited By

Zhao LWang RZou Y(2023)A hybridized weak Galerkin finite element scheme for linear elasticity problemJournal of Computational and Applied Mathematics10.1016/j.cam.2022.115024425:COnline publication date: 1-Jun-2023
https://dl.acm.org/doi/10.1016/j.cam.2022.115024
Zhou HWang XJia J(2022)Discrete maximum principle for the weak Galerkin method on triangular and rectangular meshesJournal of Computational and Applied Mathematics10.1016/j.cam.2021.113784402:COnline publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1016/j.cam.2021.113784
Meng JMei L(2020)A mixed virtual element method for the vibration problem of clamped Kirchhoff plateAdvances in Computational Mathematics10.1007/s10444-020-09810-146:5Online publication date: 10-Aug-2020
https://dl.acm.org/doi/10.1007/s10444-020-09810-1

Acceleration of Weak Galerkin Methods for the Laplacian Eigenvalue Problem
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

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Published In

cover image Journal of Scientific Computing

Journal of Scientific Computing Volume 79, Issue 2

May 2019

689 pages

ISSN:0885-7474

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Copyright © Copyright © 2019 Springer Science+Business Media, LLC, part of Springer Nature.

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Plenum Press

United States

Publication History

Published: 01 May 2019

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Cited By

Zhao LWang RZou Y(2023)A hybridized weak Galerkin finite element scheme for linear elasticity problemJournal of Computational and Applied Mathematics10.1016/j.cam.2022.115024425:COnline publication date: 1-Jun-2023
https://dl.acm.org/doi/10.1016/j.cam.2022.115024
Zhou HWang XJia J(2022)Discrete maximum principle for the weak Galerkin method on triangular and rectangular meshesJournal of Computational and Applied Mathematics10.1016/j.cam.2021.113784402:COnline publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1016/j.cam.2021.113784
Meng JMei L(2020)A mixed virtual element method for the vibration problem of clamped Kirchhoff plateAdvances in Computational Mathematics10.1007/s10444-020-09810-146:5Online publication date: 10-Aug-2020
https://dl.acm.org/doi/10.1007/s10444-020-09810-1

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