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 | References | Similar Articles | Additional Information
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 . 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
and the eigenfunctions as
in the
-seminorm, where
is the mesh-size,
depends on the smoothness of the eigenfunctions, and
results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus
for the eigenvalues and
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
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