Show Summary Details
More options …

# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2016: 0.75

Online
ISSN
1609-9389
See all formats and pricing
More options …

# A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems

Matteo Cicuttin
• Université Paris-Est, CERMICS (ENPC), 6–8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2; and Inria Paris, 2 rue Simone Iff, 75012 Paris, France
• Other articles by this author:
/ Alexandre Ern
• Université Paris-Est, CERMICS (ENPC), 6–8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2; and Inria Paris, 2 rue Simone Iff, 75012 Paris, France
• Other articles by this author:
/ Simon Lemaire
• Corresponding author
• École Polytechnique Fédérale de Lausanne, FSB-MATH-ANMC, Station 8, 1015 Lausanne, Switzerland; and Inria Lille – Nord Europe, 40 avenue Halley, 59650 Villeneuve d’Ascq, France
• Other articles by this author:
Published Online: 2018-06-19 | DOI: https://doi.org/10.1515/cmam-2018-0013

## Abstract

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation $k\ge 0$. For face unknowns that are polynomials of degree k, we devise two versions of the method, depending on the polynomial degree $\left(k-1\right)$ or k of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form $\left({\epsilon }^{\frac{1}{2}}+{H}^{k+1}+{\left(\frac{\epsilon }{H}\right)}^{\frac{1}{2}}\right)$, and we illustrate our theoretical findings on some test-cases.

MSC 2010: 65N30; 65N08; 76R50

## References

• [1]

A. Abdulle, W. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numer. 21 (2012), 1–87.

• [2]

G. Allaire, Shape Optimization by the Homogenization Method, Appl. Math. Sci. 146, Springer, New York, 2002. Google Scholar

• [3]

G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM Multiscale Model. Simul. 4 (2005), no. 3, 790–812.

• [4]

R. Araya, C. Harder, D. Paredes and F. Valentin, Multiscale hybrid-mixed method, SIAM J. Numer. Anal. 51 (2013), no. 6, 3505–3531.

• [5]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2002), no. 5, 1749–1779.

• [6]

B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. (M2AN) 50 (2016), no. 3, 879–904.

• [7]

F. Bassi, L. Botti, A. Colombo, D. A. Di Pietro and P. Tesini, On the flexibility of agglomeration-based physical space discontinuous Galerkin discretizations, J. Comput. Phys. 231 (2012), no. 1, 45–65.

• [8]

M. Bebendorf, A note on the Poincaré inequality for convex domains, Z. Anal. Anwend. 22 (2003), no. 4, 751–756. Google Scholar

• [9]

L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Meth. Appl. Sci. (M3AS) 23 (2013), 199–214.

• [10]

A. Cangiani, E. H. Georgoulis and P. Houston, hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes, Math. Models Meth. Appl. Sci. (M3AS) 24 (2014), no. 10, 2009–2041.

• [11]

E. T. Chung, S. Fu and Y. Yang, An enriched multiscale mortar space for high contrast flow problems, Commun. Comput. Phys. 23 (2018), no. 2, 476–499.

• [12]

M. Cicuttin, D. A. Di Pietro and A. Ern, Implementation of discontinuous skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming, J. Comput. Appl. Math. (2017), 10.1016/j.cam.2017.09.017. Google Scholar

• [13]

B. Cockburn, Static condensation, hybridization, and the devising of the HDG methods, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lect. Notes Comput. Sci. Eng. 114, Springer, Cham (2016), 129–177. Google Scholar

• [14]

B. Cockburn, D. A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal. (M2AN) 50 (2016), no. 3, 635–650.

• [15]

B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365.

• [16]

D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. 69, Springer, Berlin, 2012. Google Scholar

• [17]

D. A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg. 283 (2015), 1–21.

• [18]

D. A. Di Pietro and A. Ern, Hybrid high-order methods for variable-diffusion problems on general meshes, C. R. Acad. Sci. Paris Ser. I 353 (2015), 31–34.

• [19]

D. A. Di Pietro, A. Ern and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math. 14 (2014), no. 4, 461–472.

• [20]

D. A. Di Pietro, A. Ern and S. Lemaire, A review of hybrid high-order methods: Formulations, computational aspects, comparison with other methods, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations Lect. Notes Comput. Sci. Eng. 114, Springer, Cham (2016), 205–236. Google Scholar

• [21]

W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), 87–132.

• [22]

Y. Efendiev, J. Galvis and T. Y. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys. 251 (2013), 116–135.

• [23]

Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods – Theory and Applications, Surv. Tutor. Appl. Math. Sci. 4, Springer, New York, 2009. Google Scholar

• [24]

Y. Efendiev, T. Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37 (2000), no. 3, 888–910.

• [25]

Y. Efendiev, R. Lazarov, M. Moon and K. Shi, A spectral multiscale hybridizable discontinuous Galerkin method for second order elliptic problems, Comput. Methods Appl. Mech. Engrg. 292 (2015), 243–256.

• [26]

Y. Efendiev, R. Lazarov and K. Shi, A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces, SIAM J. Numer. Anal. 53 (2015), no. 1, 342–369.

• [27]

A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation, ESAIM Math. Model. Numer. Anal. (M2AN) 51 (2017), no. 4, 1367–1385. Google Scholar

• [28]

A. Ern and M. Vohralík, Stable broken ${H}^{1}$ and $H\left(\mathrm{div}\right)$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions, preprint (2016), https://hal.inria.fr/hal-01422204.

• [29]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. Google Scholar

• [30]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer, Berlin, 1986. Google Scholar

• [31]

A. Gloria, Numerical homogenization: Survey, new results, and perspectives, ESAIM Proc. 37 (2012), 50–116.

• [32]

P. Henning and D. Peterseim, Oversampling for the multiscale finite element method, SIAM Multiscale Model. Simul. 11 (2013), no. 4, 1149–1175.

• [33]

J. S. Hesthaven, S. Zhang and X. Zhu, High-order multiscale finite element method for elliptic problems, SIAM Multiscale Model. Simul. 12 (2014), no. 2, 650–666.

• [34]

T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), 169–189.

• [35]

T. Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68 (1999), no. 227, 913–943.

• [36]

T. Y. Hou, X.-H. Wu and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov–Galerkin formulation, Commun. Math. Sci. 2 (2004), no. 2, 185–205.

• [37]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994. Google Scholar

• [38]

A. Konaté, Méthode multi-échelle pour la simulation d’écoulements miscibles en milieux poreux, Ph.D. thesis, Université Pierre et Marie Curie, 2017, https://tel.archives-ouvertes.fr/tel-01558994.

• [39]

C. Le Bris, F. Legoll and A. Lozinski, MsFEM à la Crouzeix–Raviart for highly oscillatory elliptic problems, Chin. Ann. Math. Ser. B 34 (2013), no. 1, 113–138.

• [40]

C. Le Bris, F. Legoll and A. Lozinski, An MsFEM-type approach for perforated domains, SIAM Multiscale Model. Simul. 12 (2014), no. 3, 1046–1077.

• [41]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comp. 83 (2014), 2583–2603.

• [42]

L. Mu, J. Wang and X. Ye, A weak Galerkin generalized multiscale finite element method, J. Comput. Appl. Math. 305 (2016), 68–81.

• [43]

D. Paredes, F. Valentin and H. M. Versieux, On the robustness of multiscale hybrid-mixed methods, Math. Comp. 86 (2017), 525–548. Google Scholar

• [44]

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2045–2066.

• [45]

A. Veeser and R. Verfürth, Poincaré constants for finite element stars, IMA J. Numer. Anal. 32 (2012), no. 1, 30–47.

• [46]

E. L. Wachspress, A Rational Finite Element Basis, Math. Sci. Eng. 114, Academic Press, New York, 1975. Google Scholar

• [47]

J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math. 241 (2013), 103–115.

Accepted: 2018-06-01

Published Online: 2018-06-19

Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.