The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us to perform a complete and generic study of the well-known (but rarely well-studied) mass lumping process. They also allow an easy check of the mathematical properties of new schemes, by developing a generic process for eliminating unknowns via barycentric condensation, and by designing a concept of discrete functional analysis toolbox for schemes based on polytopal meshes.
DOI : 10.1051/m2an/2015079
Mots-clés : Gradient scheme, gradient discretisation, numerical scheme, diffusion equations, convergence analysis, discrete functional analysis
@article{M2AN_2016__50_3_749_0,
author = {Droniou, J\'erome and Eymard, Robert and Herbin, Rapha\`ele},
title = {Gradient schemes: {Generic} tools for the numerical analysis of diffusion equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {749--781},
publisher = {EDP-Sciences},
volume = {50},
number = {3},
year = {2016},
doi = {10.1051/m2an/2015079},
zbl = {1346.65042},
mrnumber = {3507272},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015079/}
}
TY - JOUR AU - Droniou, Jérome AU - Eymard, Robert AU - Herbin, Raphaèle TI - Gradient schemes: Generic tools for the numerical analysis of diffusion equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 749 EP - 781 VL - 50 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015079/ DO - 10.1051/m2an/2015079 LA - en ID - M2AN_2016__50_3_749_0 ER -
%0 Journal Article %A Droniou, Jérome %A Eymard, Robert %A Herbin, Raphaèle %T Gradient schemes: Generic tools for the numerical analysis of diffusion equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 749-781 %V 50 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015079/ %R 10.1051/m2an/2015079 %G en %F M2AN_2016__50_3_749_0
Droniou, Jérome; Eymard, Robert; Herbin, Raphaèle. Gradient schemes: Generic tools for the numerical analysis of diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 749-781. doi : 10.1051/m2an/2015079. http://archive.numdam.org/articles/10.1051/m2an/2015079/
, , and , Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127 (1996) 2–14. | DOI | Zbl
, and , Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations. J. Hyperbolic Differ. Equ. 7 (2010) 1–67. | DOI | MR | Zbl
, and , On 3D DDFV discretization of gradient and divergence operators: discrete functional analysis tools and applications to degenerate parabolic problems. Comput. Methods Appl. Math. 13 (2013) 369–410. | DOI | MR | Zbl
B. Andreianov, M. Bendahmane and K. Karlsen, A Gradient Reconstruction Formula for Finite-volume Schemes and Discrete Duality. In Proc. of Finite Volumes for Complex Applications V. ISTE, London (2008) 161–168. | MR
Y. Alnashri and J. Droniou, Gradient schemes for variational inequalities (2014). Submitted.
, and , Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Eqs. 23 (2007) 145–195. | DOI | MR | Zbl
B. Andreianov and F. Hubert, Personal communication (2015).
and , Mixed and conforming finite element methods; implementation, postprocessing and error estimates. Model. Math. Anal. Num. 19 (1985) 7–32. | DOI | Numdam | MR | Zbl
, and , Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Amer. Math. Soc. (N.S.) 47 (2010) 281–354. | DOI | MR | Zbl
J. Bonelle, Compatible Discrete Operator schemes on polyhedral meshes for elliptic and Stokes equations. Ph.D. thesis, University of Paris-Est (2014).
and , Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN 48 (2014) 553–581. | DOI | Numdam | MR | Zbl
and , Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal. 46 (2008) 3032–3070. | DOI | MR | Zbl
K. Brenner, M. Groza, C. Guichard, G. Lebeau and R. Masson, Gradient discretization of hybrid dimensional darcy flows in fractured porous media. To appear in: Numer. Math. (2015). Doi: | DOI | MR
, and , Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. | DOI | Numdam | MR | Zbl
C. Cancès and C. Guichard, Numerical analysis of a robust entropy-diminishing Finite Volume scheme for parabolic equations with gradient structure (2015). | MR
P.G. Ciarlet, The finite element method for elliptic problems. Access Online via Elsevier (1978). | MR | Zbl
and , A 3d discrete duality finite volume method for nonlinear elliptic equations. SIAM J. Sci. Comp. 33 (2011) 1739–1764. | DOI | MR | Zbl
, , and , A 2D/3D discrete duality finite volume scheme. Application to ECG simulation. Int. J. Finite 6 (2009) 24. | MR | Zbl
Y. Coudière, F. Hubert and G. Manzini, A CeVeFE DDFV scheme for discontinuous anisotropic permeability tensors. In Proc. of Finite volumes for complex applications VI. Vol. 4 of Springer Proc. Math. Springer, Heidelberg (2011) 283–291. | MR | Zbl
Y. Coudière, F. Hubert and G. Manzini, A CeVeFE DDFV Scheme for Discontinuous Anisotropic Permeability Tensors. In Finite Volumes for Complex Applications. VI. Problems & Perspectives. Vol. 4 of Springer Proc. Math. Springer, Heidelberg (2011) 283–291. | MR | Zbl
D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69. Springer (2012). | MR | Zbl
and , A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. | DOI | Numdam | MR | Zbl
, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Special issue on Recent Techniques for PDE Discretizations on Polyhedral Meshes. M3AS: Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. | 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 , Gradient schemes for linear and non-linear elasticity equations. Numer. Math. 129 (2015) 251–277. | DOI | MR | Zbl
and , Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations. Numer. Math. 132 (2016) 721–766. | DOI | MR | Zbl
, , and , A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20 (2010) 265–295. | DOI | MR | Zbl
, , and , Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. M3AS: Math. Models Methods Appl. Sci. 23 (2013) 2395–2432. | MR | Zbl
J. Droniou, R. Eymard and P. Féron, Gradient schemes for Stokes problem. To appear in IMA J. Numer. Anal. (2015). Doi: | DOI | MR
J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, Gradient schemes for elliptic and parabolic problems. In preparation (2015).
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. In Techniques of Scientific Computing, Part III, edited by P.G. Ciarlet and J.-L. Lions. Handbook of Numerical Analysis VII. North-Holland, Amsterdam (2000) 713–1020. | MR | Zbl
, , and , Analysis tools for finite volume schemes. Acta Math. Univ. Comenian. (N.S.) 76 (2007) 111–136. | 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. Eymard, C. Guichard, R. Herbin and R. Masson, Vertex Centred Discretization of Two-phase Darcy Flows on General Meshes. In Congrès National de Mathématiques Appliquées et Industrielles, vol. 35 of ESAIM Proc. EDP Sci., Les Ulis (2011) 59–78. | MR | Zbl
R. Eymard, G. Henry, R. Herbin, F. Hubert, R. Klofkorn and G. Manzini, 3d Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids. In Proc. of Finite Volumes for Complex Applications VI, Praha. Springer, Springer (2011) 895–930. | Zbl
, and . Small-stencil 3d schemes for diffusive flows in porous media. ESAIM: M2AN 46 (2012) 265–290. | DOI | Numdam | MR | Zbl
R. Eymard, C. Guichard, R. Herbin and R. Masson, Vertex-centred discretization of multiphase compositional darcy flows on general meshes. Comput. Geosci. (2012) 1–19. | MR
, , Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes. Communic. Pure and Appl. Anal. 11 (2012) 147–172. | MR | Zbl
R. Eymard, P. Feron, T. Gallouët, R. Herbin and C. Guichard, Gradient schemes for the Stefan problem. Int. J. Finite Volumes, 10s (2013). | MR
, , and , Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. ZAMM Z. Angew. Math. Mech. 94 (2014) 560–585. | DOI | MR | Zbl
, and , RTk mixed finite elements for some nonlinear problems. Math. Comput. Simul. 118 (2015) 186–197. | DOI | MR | Zbl
, , , and , Applications of approximate gradient schemes for nonlinear parabolic equations. Appl. Math. 60 (2015) 135–156. | DOI | MR | Zbl
I. Faille, Modélisation bidimensionnelle de la genèse et de la migration des hydrocarbures dans un bassin sédimentaire. Ph.D. thesis, University Joseph Fourier – Grenoble I (1992).
, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192 (2003) 1939–1959. | DOI | MR | Zbl
, Approximation of 2-d and 3-d diffusion operators with variable full tensor coefficients on arbitrary meshes. Comput. Methods Appl. Mech. Eng. 196 (2007) 2497–2526. | DOI | MR | Zbl
, A finite volume method for approximating 3d diffusion operators on general meshes. J. Comput. Phys. 228 (2009) 5763–5786. | DOI | MR | Zbl
, and , Mimetic finite difference method. J. Comput. Phys. 257 (2011) 1163–1227. | DOI | MR | Zbl
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