Spectral approximations by the HDG method

Gopalakrishnan, J.; Li, F.; Nguyen, N.-C.; Peraire, J.

doi:10.1090/S0025-5718-2014-02885-8

Spectral approximations by the HDG method
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by J. Gopalakrishnan, F. Li, N.-C. Nguyen and J. Peraire PDF

Math. Comp. 84 (2015), 1037-1059 Request permission

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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