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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
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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.


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Additional Information

J. Gopalakrishnan
Affiliation: Portland State University, P.O. Box 751, Portland, Oregon 97207-0751
Email: gjay@pdx.edu

F. Li
Affiliation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180
Email: lif@rpi.edu

N.-C. Nguyen
Affiliation: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: cuongng@mit.edu

J. Peraire
Affiliation: Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: peraire@mit.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02885-8
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