We build a bridge between the hybrid high-order (HHO) and the hybridizable discontinuous Galerkin (HDG) methods in the setting of a model diffusion problem. First, we briefly recall the construction of HHO methods and derive some new variants. Then, by casting the HHO method in mixed form, we identify the numerical flux so that the HHO method can be compared to HDG methods. In turn, the incorporation of the HHO method into the HDG framework brings up new, efficient choices of the local spaces and a new, subtle construction of the numerical flux ensuring optimal orders of convergence on meshes made of general shape-regular polyhedral elements. Numerical experiments comparing two of these methods are shown.
DOI : 10.1051/m2an/2015051
Mots clés : Hybridizable discontinuous Galerkin, hybrid high-order, variable diffusion problems
@article{M2AN_2016__50_3_635_0,
author = {Cockburn, Bernardo and Di Pietro, Daniele A. and Ern, Alexandre},
title = {Bridging the hybrid high-order and hybridizable discontinuous {Galerkin} methods},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {635--650},
publisher = {EDP-Sciences},
volume = {50},
number = {3},
year = {2016},
doi = {10.1051/m2an/2015051},
zbl = {1341.65045},
mrnumber = {3507267},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015051/}
}
TY - JOUR AU - Cockburn, Bernardo AU - Di Pietro, Daniele A. AU - Ern, Alexandre TI - Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 635 EP - 650 VL - 50 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015051/ DO - 10.1051/m2an/2015051 LA - en ID - M2AN_2016__50_3_635_0 ER -
%0 Journal Article %A Cockburn, Bernardo %A Di Pietro, Daniele A. %A Ern, Alexandre %T Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 635-650 %V 50 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015051/ %R 10.1051/m2an/2015051 %G en %F M2AN_2016__50_3_635_0
Cockburn, Bernardo; Di Pietro, Daniele A.; Ern, Alexandre. Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 3, pp. 635-650. doi : 10.1051/m2an/2015051. http://archive.numdam.org/articles/10.1051/m2an/2015051/
, and , Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34 (1997) 828–852. | DOI | MR | Zbl
B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. Preprint (2014). | arXiv | Numdam | MR
L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. MS&A. Springer (2014). | MR | Zbl
and , Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN 48 (2014) 553–581. | DOI | Numdam | MR | Zbl
, and , Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. | DOI | MR | Zbl
, , and , A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3682–3692. | DOI | MR | Zbl
, , and , An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676–1706. | DOI | MR | Zbl
, BDM mixed methods for a nonlinear elliptic problem. J. Comput. Appl. Math. 53 (1994) 207–223. | DOI | MR | Zbl
, and , Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. | DOI | MR | Zbl
, and , A projection-based error analysis of HDG methods. Math. Comput. 79 (2010) 1351–1367. | DOI | MR | Zbl
, and , Conditions for superconvergence of HDG methods for second-order elliptic problems. Math. Comput. 81 (2012) 1327–1353. | DOI | MR | Zbl
and , Devising HDG methods for Stokes flow: An overview. Comput. Fluids 98 (2014) 221–229. | DOI | MR | Zbl
, and , A discontinuous-skeletal method for advection-diffusion-reaction on general meshes. SIAM J. Numer. Anal. 53 (2015) 2135–2157. | DOI | MR | Zbl
D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Math. Appl. Springer-Verlag, Berlin (2012). | MR | Zbl
and , Equilibrated tractions for the Hybrid High-Order method. C. R. Acad. Sci Paris, Ser. I 353 (2015) 279–282. | DOI | MR | Zbl
, and , A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Engrg. 283 2015 1–21. | DOI | MR | Zbl
, and , Hybrid high-order methods for variable-diffusion problems on general meshes. C. R. Acad. Sci Paris, Ser. I 353 (2015) 31–34. | DOI | MR | Zbl
, and . An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Meth. Appl. Math. 14 (2014) 461–472. | DOI | MR | Zbl
and , An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84 (2015) 1–31. | DOI | MR | Zbl
and , A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 35–71. | DOI | MR | Zbl
, , and , A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. M3AS Math. Models Methods Appl. Sci. 20 (2010) 1–31. | MR | Zbl
and , Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441–463. | DOI | MR | Zbl
, and , Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043. | DOI | MR | Zbl
R. Herbin and F. Hubert, Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. In Finite Volumes for Complex Applications V. Edited by R. Eymard and J.-M. Hérard. John Wiley and Sons (2008) 659–692. | MR
, and , A hybridizable discontinuous Galerkin formulation for nonlinear elasticity. Comput. Methods Appl. Mech. Engrg. 283 (2015) 303–329. | DOI | MR | Zbl
, A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors. Numer. Methods Partial Differ. Equ. 9 (1993) 339–355. | DOI | MR | Zbl
, and , Mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci. 8 (2004) 301–324. | DOI | MR | Zbl
C. Le Potier, A finite Volume Method for the Approximation of Highly Anisotropic Diffusion Operators on Unstructured Meshes. In Finite Volumes for Complex Applications IV (2005). | MR
C. Lehrenfeld, Hybrid Discontinuous Galerkin methods for incompressible flow problems. Diploma thesis, MathCCES/IGPM, RWTH Aachen (2010).
and . A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys. 272 (2014) 360–385. | DOI | MR | Zbl
, A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. 65 (2015) 327–340. | DOI | MR | Zbl
W. Qiu and K. Shi, An HDG method for linear elasticity with strongly symmetric stresses. Preprint (2015). | arXiv
and . An HDG method for convection-diffusion equations. J. Sci. Comput. 66 (2016) 346–357. | DOI | MR | Zbl
S.-C. Soon, Hybridizable discontinuous Galerkin methods for solid mechanics. Ph.D. thesis, University of Minnesota, Minneapolis (2008).
, and , A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Engrg. 80 (2009) 1058–1092. | DOI | MR | Zbl
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