Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


Spectral approximations by the HDG method

Authors: J. Gopalakrishnan, F. Li, N.-C. Nguyen and J. Peraire
Journal: Math. Comp. 84 (2015), 1037-1059
MSC (2010): Primary 65N12, 65N25, 65N30
Published electronically: December 30, 2014
MathSciNet review: 3315499
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the numerical approximation of the spectrum of a second-order elliptic eigenvalue problem by the hybridizable discontinuous Galerkin (HDG) method. We show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate $ 2k+1$ and $ k+1$, respectively. Here $ k$ is the degree of the polynomials used to approximate the solution, its flux, and the numerical traces. Our numerical studies show that a Rayleigh quotient-like formula applied to certain locally postprocessed approximations can yield eigenvalues that converge faster at the rate $ 2k + 2$ for the HDG method as well as for the Brezzi-Douglas-Marini (BDM) method. We also derive and study a condensed nonlinear eigenproblem for the numerical traces obtained by eliminating all the other variables.

References [Enhancements On Off] (What's this?)

  • [1] Philip M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR 0443383 (56 #1753)
  • [2] Paola F. Antonietti, Annalisa Buffa, and Ilaria Perugia, Discontinuous Galerkin approximation of the Laplace eigenproblem, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 25-28, 3483-3503. MR 2220929 (2006m:65249),
  • [3] F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math. 2 (1960), 137-141. MR 0118729 (22 #9500)
  • [4] Timo Betcke and Lloyd N. Trefethen, Reviving the method of particular solutions, SIAM Rev. 47 (2005), no. 3, 469-491 (electronic). MR 2178637 (2006k:65344),
  • [5] Daniele Boffi, Franco Brezzi, and Lucia Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp. 69 (2000), no. 229, 121-140. MR 1642801 (2000i:65175),
  • [6] J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525-549. MR 0366029 (51 #2280)
  • [7] B. Cockburn, O. Dubois, J. Gopalakrishnan, and S. Tan, Multigrid for an HDG method, IMA J. Numer. Anal. 34 (2014), no. 4, 1386-1425. MR 3269430,
  • [8] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319-1365. MR 2485455 (2010b:65251),
  • [9] B. Cockburn, J. Gopalakrishnan, F. Li, N.-C. Nguyen, and J. Peraire, Hybridization and postprocessing techniques for mixed eigenfunctions, SIAM J. Numer. Anal. 48 (2010), no. 3, 857-881. MR 2669393 (2011f:65244),
  • [10] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Francisco-Javier Sayas, A projection-based error analysis of HDG methods, Math. Comp. 79 (2010), no. 271, 1351-1367. MR 2629996 (2011d:65354),
  • [11] Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. I. The problem of convergence, RAIRO Anal. Numér. 12 (1978), no. 2, 97-112, iii (English, with French summary). MR 0483400 (58 #3404a)
  • [12] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988; A Wiley-Interscience Publication. MR 1009163 (90g:47001b)
  • [13] Stefano Giani and Edward J. C. Hall, An a posteriori error estimator for $ hp$-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems, Math. Models Methods Appl. Sci. 22 (2012), no. 10, 1250030, 35. MR 2974168,
  • [14] Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. MR 0107819 (21 #6541)
  • [15] height 2pt depth -1.6pt width 23pt, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
    Reprint of the 1980 edition.
  • [16] B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), no. 154, 427-453. MR 606505 (82b:65108),
  • [17] John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712-725. MR 0383117 (52 #3998)
  • [18] Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151-167 (English, with French summary). MR 1086845 (92a:65303)
  • [19] Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441-479 (German). MR 1511670,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N12, 65N25, 65N30

Retrieve articles in all journals with MSC (2010): 65N12, 65N25, 65N30

Additional Information

J. Gopalakrishnan
Affiliation: Portland State University, P.O. Box 751, Portland, Oregon 97207-0751

F. Li
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180

N.-C. Nguyen
Affiliation: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

J. Peraire
Affiliation: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Keywords: HDG, nonlinear, eigenvalue, eigenfunction, BDM, postprocessing, condensation, hybridization, pollution, perturbation
Received by editor(s): July 3, 2012
Received by editor(s) in revised form: July 11, 2013
Published electronically: December 30, 2014
Additional Notes: This work was partially supported by NSF through grants DMS-1211635, DMS-1318916, and the CAREER award DMS-0847241, by an Alfred P. Sloan Research Fellowship, and by AFOSR under grant FA9550-12-0357.
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society