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Spectral approximation of elliptic operators by the Hybrid High-Order method


Authors: Victor Calo, Matteo Cicuttin, Quanling Deng and Alexandre Ern
Journal: Math. Comp. 88 (2019), 1559-1586
MSC (2010): Primary 65N15, 65N30, 65N35, 35J05
DOI: https://doi.org/10.1090/mcom/3405
Published electronically: December 20, 2018
MathSciNet review: 3925477
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Abstract: We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $ k\geq 0$. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $ h^{2t}$ and the eigenfunctions as $ h^{t}$ in the $ H^1$-seminorm, where $ h$ is the mesh-size, $ t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $ s>\frac 12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $ h^{2k+2}$ for the eigenvalues and $ h^{k+1}$ for the eigenfunctions. Our theoretical findings, which improve recent error estimates for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension for smooth eigenfunctions that the eigenvalues superconverge as $ h^{2k+4}$ for a specific value of the stabilization parameter.


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Additional Information

Victor Calo
Affiliation: Department of Applied Geology, Western Australian School of Mines, Curtin University, Kent Street, Bentley, Perth, WA 6102, Australia; and Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Kensington, Perth, WA 6152, Australia
Email: victor.calo@curtin.edu.au

Matteo Cicuttin
Affiliation: University Paris-Est, CERMICS (ENPC), 77455 Marne la Vallée cedex 2, and INRIA Paris, 75589 Paris, France
Email: matteo.cicuttin@enpc.fr

Quanling Deng
Affiliation: Curtin Institute for Computation and Department of Applied Geology, Western Australian School of Mines, Curtin University, Kent Street, Bentley, Perth, WA 6102, Australia
Email: quanling.deng@curtin.edu.au

Alexandre Ern
Affiliation: University Paris-Est, CERMICS (ENPC), 77455 Marne la Vallée cedex 2, and INRIA Paris, 75589 Paris, France
Email: alexandre.ern@enpc.fr

DOI: https://doi.org/10.1090/mcom/3405
Keywords: Hybrid high-order methods, eigenvalue approximation, eigenfunction approximation, spectrum analysis, error analysis
Received by editor(s): November 3, 2017
Received by editor(s) in revised form: July 19, 2018, and August 7, 2018
Published electronically: December 20, 2018
Additional Notes: This article was supported in part by the European Union’s Horizon 2020 Research and Innovation Program of the Marie Skłodowska-Curie grant agreement No. 777778.
The third author is the corresponding author.
Article copyright: © Copyright 2018 American Mathematical Society