Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 2, pp. 553-581.

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L2-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

DOI : https://doi.org/10.1051/m2an/2013104
Classification : 65N12,  65N08,  65N30
Mots clés : compatible schemes, mimetic discretization, Hodge operator, error analysis, elliptic problems, polyhedral meshes
@article{M2AN_2014__48_2_553_0,
author = {Bonelle, J\'er\^ome and Ern, Alexandre},
title = {Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {553--581},
publisher = {EDP-Sciences},
volume = {48},
number = {2},
year = {2014},
doi = {10.1051/m2an/2013104},
zbl = {1297.65132},
mrnumber = {3177857},
language = {en},
url = {archive.numdam.org/item/M2AN_2014__48_2_553_0/}
}
Bonelle, Jérôme; Ern, Alexandre. Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) no. 2, pp. 553-581. doi : 10.1051/m2an/2013104. http://archive.numdam.org/item/M2AN_2014__48_2_553_0/

[1] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Meth. Appl. Sci. 21 (1998) 823-864. | MR 1626990 | Zbl 0914.35094

[2] B. Andreianov, F. Boyer and F. Hubert, Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Eqs. 23 (2007) 145-195. | MR 2275464 | Zbl 1111.65101

[3] D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15 (2006) 1-155. | MR 2269741 | Zbl 1185.65204

[4] A. Back, Étude théorique et numérique des équations de Vlasov-Maxwell dans le formalisme covariant. Ph.D. thesis, University of Strasbourg (2011).

[5] P. Bochev and J.M. Hyman, Principles of mimetic discretizations of differential operators, Compatible Spatial Discretization. In vol. 142 of The IMA Volumes Math. Appl., edited by D. Arnold, P. Bochev, R. Lehoucq, R.A. Nicolaides and M. Shashkov (2005) 89-120. | MR 2249347 | Zbl 1110.65103

[6] A. Bossavit, On the geometry of electromagnetism. J. Japan Soc. Appl. Electromagn. Mech. 6 (1998) (no 1) 17-28, (no 2) 114-23, (no 3) 233-40, (no 4) 318-26.

[7] A. Bossavit, Computational electromagnetism and geometry. J. Japan Soc. Appl. Electromagn. Mech. 7-8 (1999-2000) (no 1) 150-9, (no 2) 294-301, (no 3) 401-8, (no 4) 102-9, (no 5) 203-9, (no 6) 372-7.

[8] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite difference for elliptic problem. Math. Model. Numer. Anal. 43 (2009) 277-295. | Numdam | MR 2512497 | Zbl 1177.65164

[9] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872-1896. | MR 2192322 | Zbl 1108.65102

[10] A. Buffa and S.H. Christiansen, A dual finite element complex on the barycentric refinement. Math. Comput. 76 (2007) 1743-1769. | MR 2336266 | Zbl 1130.65108

[11] S.H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes. Math. Models Methods Appl. Sci. 18 (2008) 739-757. | MR 2413036 | Zbl 1153.65005

[12] S.H. Christiansen, H.Z. Munthe-Kaas and B. Owren, Topics in structure-preserving discretization. Acta Numer. 20 (2011) 1-119. | MR 2805152 | Zbl 1233.65087

[13] L. Codecasa, R. Specogna and F. Trevisan, Base functions and discrete constitutive relations for staggered polyhedral grids. Comput. Methods Appl. Mech. Engrg. 198 (2009) 1117-1123. | MR 2498867 | Zbl 1229.78025

[14] L. Codecasa, R. Specogna and F. Trevisan, A new set of basis functions for the discrete geometric approach. J. Comput. Phys. 229 (2010) 7401-7410. | MR 2677785 | Zbl 1196.78027

[15] L. Codecasa and F. Trevisan, Convergence of electromagnetic problems modelled by discrete geometric approach. CMES 58 (2010) 15-44. | MR 2674759 | Zbl 1231.78046

[16] M. Desbrun, A.N. Hirani, M. Leok and J.E. Marsden, Discrete Exterior Calculus. Technical report (2005).

[17] D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, in vol. 69 of SMAI Math. Appl. Springer (2012). | MR 2882148 | Zbl 1231.65209

[18] J. Dodziuk, Finite-difference approach to the Hodge theory of harmonic forms. Amer. J. Math. 98 (1976) 79-104. | MR 407872 | Zbl 0324.58001

[19] K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203-1249. | Numdam | MR 2195910 | Zbl 1086.65108

[20] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105 (2006) 35-71. | MR 2257385 | Zbl 1109.65099

[21] J. Droniou, R. Eymard, T. Gallouët and R. Herbin, A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods. Math. Models and Methods Appl. Sci. 20 (2010) 265-295. | MR 2649153 | Zbl 1191.65142

[22] R. Eymard, T. Gallouët and R. Herbin, 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. | MR 2727814 | Zbl 1202.65144

[23] R. Eymard, C. Guichard and R. Herbin, Small stencil 3d schemes for diffusive flows in porious media. ESAIM: M2AN 46 (2012) 265-290. | Numdam | MR 2855643 | Zbl 1271.76324

[24] R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klöfkorn and G. Manzini, 3d benchmark on discretization schemes for anisotropic diffusion problems on general grids, in vol. 2 of Finite Volumes for Complex Applic. VI - Problems Perspectives. Springer (2011) 95-130. | MR 2882736 | Zbl pre06116001

[25] A. Gillette, Stability of dual discretization methods for partial differential equations. Ph.D. thesis, University of Texas at Austin (2011).

[26] R. Hiptmair, Discrete hodge operators: An algebraic perspective. Progress In Electromagnetics Research 32 (2001) 247-269. | MR 1872728 | Zbl 0993.65130

[27] Xiao Hua Hu and R.A. Nicolaides, Covolume techniques for anisotropic media. Numer. Math. 61 (1992) 215-234. | MR 1147577 | Zbl 0734.65088

[28] J. Hyman and J. Scovel, Deriving mimetic difference approximations to differential operators using algebraic topology. Los Alamos National Laboratory (1988).

[29] J. Kreeft, A. Palha and M. Gerritsma, Mimetic framework on curvilinear quadrilaterals of arbitrary order. Technical Report, Delft University (2011) ArXiv: 1111.4304v1.

[30] C. Mattiussi, The finite volume, finite element, and finite difference methods as numerical methods for physical field problems. In vol. 113 of Advances in Imaging and Electron Phys. Elsevier (2000) 1-146.

[31] J.B. Perot and V. Subramanian, Discrete calculus methods for diffusion. J. Comput. Phys. 224 (2007) 59-81. | MR 2322260 | Zbl 1120.65325

[32] T. Tarhasaari, L. Kettunen and A. Bossavit, Some realizations of a discrete hodge operator: A reinterpretation of finite element techniques. IEEE Transactions on Magnetics 35 (1999) 1494-1497.

[33] E. Tonti, On the formal structure of physical theories. Instituto di matematica, Politecnico, Milano (1975).

[34] M. Vohralík and B. Wohlmuth, Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. Math. Models Methods Appl. Sci. 23 (2013) 803-838. | MR 3028542 | Zbl 1264.65198

[35] H. Whitney, Geometric integration theory. Princeton University Press, Princeton, N.J. (1957). | MR 87148 | Zbl 0083.28204

[36] S. Zaglmayr, High order finite element methods for electromagnetic field computation. Ph.D. thesis, Johannes Kepler Universität Linz (2006).