The arbitrary order mixed mimetic finite difference method for the diffusion equation

Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco

doi:10.1051/m2an/2015088

Gyrya, Vitaliy ¹ ; Lipnikov, Konstantin ¹ ; Manzini, Gianmarco ^1,^2,³

¹ Los Alamos National Laboratory, Theoretical Division, Group T-5, MS B284, Los Alamos, NM-87545, USA
² Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche (IMATI-CNR), via Ferrata 1, 27100 Pavia, Italy
³ Centro di Simulazione Numerica Avanzata (CeSNA) – IUSS Pavia, v.le Lungo Ticino Sforza 56, 27100 Pavia, Italy

ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 851-877.

Résumé

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux and scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.

Reçu le : 2015-04-19

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/m2an/2015088

Classification : 65N30, 65N12, 65G99, 76R99
Mots-clés : Mimetic finite difference method, polygonal mesh, high-order discretization, Poisson problem, mixed formulation

Affiliations des auteurs :

Gyrya, Vitaliy ¹ ; Lipnikov, Konstantin ¹ ; Manzini, Gianmarco ^{1, 2, 3}

@article{M2AN_2016__50_3_851_0,
     author = {Gyrya, Vitaliy and Lipnikov, Konstantin and Manzini, Gianmarco},
     title = {The arbitrary order mixed mimetic finite difference method for the diffusion equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {851--877},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {3},
     year = {2016},
     doi = {10.1051/m2an/2015088},
     mrnumber = {3507276},
     zbl = {1342.65202},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2015088/}
}

TY  - JOUR
AU  - Gyrya, Vitaliy
AU  - Lipnikov, Konstantin
AU  - Manzini, Gianmarco
TI  - The arbitrary order mixed mimetic finite difference method for the diffusion equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 851
EP  - 877
VL  - 50
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2015088/
DO  - 10.1051/m2an/2015088
LA  - en
ID  - M2AN_2016__50_3_851_0
ER  -

%0 Journal Article
%A Gyrya, Vitaliy
%A Lipnikov, Konstantin
%A Manzini, Gianmarco
%T The arbitrary order mixed mimetic finite difference method for the diffusion equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 851-877
%V 50
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2015088/
%R 10.1051/m2an/2015088
%G en
%F M2AN_2016__50_3_851_0

Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco. The arbitrary order mixed mimetic finite difference method for the diffusion equation. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 851-877. doi : 10.1051/m2an/2015088. http://archive.numdam.org/articles/10.1051/m2an/2015088/

Bibliographie
Cité par

I. Aavatsmark, T. Barkve, O. Bœ and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part i: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. | DOI | MR | Zbl

I. Aavatsmark, T. Barkve, O. Bœ and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part ii: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717–1736. | DOI | MR | Zbl

P. Antonietti, L. Beirão Da Veiga, N. Bigoni and M. Verani, Mimetic finite differences for nonlinear and control problems. Math. Models Methods Appl. Sci. 24 (2014) 1457–1493. | DOI | MR | Zbl

L. Beirão Da Veiga, A mimetic discretization method for linear elasticity. ESAIM: M2AN 44 (2010) 231–250. | DOI | Numdam | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 119–214. | DOI | MR | Zbl

L. Beirão Da Veiga, J. Droniou and G. Manzini, A unified approach to handle convection term in finite volumes and mimetic discretization methods for elliptic problems. IMA J. Num. Anal. 31 (2011) 1357–1401. | DOI | MR | Zbl

L. Beirão Da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys. 228 (2009) 7215–7232. | DOI | MR | Zbl

L. Beirão Da Veiga, K. Lipnikov and G. Manzini, Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325–356. | DOI | MR | Zbl

L. Beirão Da Veiga, K. Lipnikov and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. | DOI | MR | Zbl

L. Beirão Da Veiga, K. Lipnikov and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48 (2011) 1419–1443. | DOI | MR | Zbl

L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method. Vol. 11 of Model. Simul. Appl. 1st edition. Springer-Verlag, New York (2014). | MR | Zbl

L. Beirão Da Veiga, and G. Manzini, A higher-order formulation of the mimetic finite difference method. SIAM J. Sci. Comput. 31 (2008) 732–760. | DOI | MR | Zbl

L. Beirão Da Veiga, and D. Mora, A mimetic discretization of the Reissner–Mindlin plate bending problem. Numer. Math. 117 (2011) 425–462. | DOI | MR | Zbl

P. Bochev and J.M. Hyman, Principle of mimetic discretizations of differential operators. Compatible discretizations. In Proc. of IMA hot topics workshop on compatible discretizations, edited by D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov. IMA. Springer-Verlag 142 (2006) 89–120. | MR | Zbl

D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Springer Series Comput. Math. Springer, Berlin, Heidelberg (2013). | MR | Zbl

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin/Heidelberg (1994). | MR | Zbl

F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. | DOI | Numdam | MR | Zbl

F. Brezzi, A. Buffa and G. Manzini, Mimetic inner products for discrete differential forms. J. Comput. Phys. B 257 (2014) 1228–1259. | DOI | MR | Zbl

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. | DOI | MR | Zbl

J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739–765. | DOI | MR | Zbl

A. Cangiani, F. Gardini and G. Manzini, Convergence of the mimetic finite difference method for eigenvalue problems in mixed form. Comput. Methods Appl. Mech. Engrg. 200 (2011) 1150–1160. | DOI | MR | Zbl

A. Cangiani and G. Manzini, Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Methods Appl. Mech. Engrg. 197 (2008) 933–945. | DOI | MR | Zbl

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) 1319–1365. | DOI | MR | Zbl

Y. Coudière and G. Manzini, The discrete duality finite volume method for convection-diffusion problems. SIAM J. Numer. Anal. 47 (2010) 4163–4192. | DOI | MR | Zbl

D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications. Springer (2011). | MR | Zbl

D.A. Di Pietro and A. Ern, Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes, hal-00918482-v3 (2013).

D.A. Di Pietro and A. Ern, Hybrid high-order methods for variable diffusion problems on general meshes. C. R. Math. 353 (2014) 31–34. | DOI | MR | Zbl

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. | DOI | Numdam | MR | Zbl

J. Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. | DOI | MR | Zbl

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 Methods Appl. Sci. 20 (2010) 265–295. | DOI | MR | Zbl

J. Droniou, R. Eymard, T. Gallouët and R. Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. 23 (2013) 2395–2432. | DOI | MR | Zbl

R. Eymard, T. Gallouët and R. Herbin, The finite volume method. In Handbook for Numerical Analysis, edited by P. Ciarlet and J.L. Lions. North Holland (2000) 715–1022. | MR | Zbl

R. Eymard, T. Gallouet and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilization and hybrid interface. IMA J. Numer. Anal. 30 (2010) 1009–1043. | DOI | MR | Zbl

V. Gyrya and K. Lipnikov, High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227 (2008) 8841–8854. | DOI | MR | Zbl

F. Hermeline, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Engrg. 192 (2003) 1939–1959. | DOI | MR | Zbl

J. Hyman and M. Shashkov, Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion. Progress Electromagn. Res. 32 (2001) 89–121. | DOI

J.M. Hyman, M.J. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130–148. | DOI | MR | Zbl

G. Lin, J. Liu and F. Sadre-Marandi, A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods. J. Comput. Appl. Math. 273 (2015) 346–362. | DOI | MR | Zbl

K. Lipnikov and G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys. 227 (2014) 360–385. | DOI | MR | Zbl

K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa, The mimetic finite difference method for 3D magnetostatics fields problems. J. Comput. Phys. 230 (2011) 305–328. | DOI | MR | Zbl

K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. B 257 (2014) 1163–1227. | DOI | MR | Zbl

K. Lipnikov, G. Manzini and D. Svyatskiy, Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230 (2011) 2620–2642. | DOI | MR | Zbl

K.N. Lipnikov, J.D. Moulton, G. Manzini and M.J. Shashkov, The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient. Technical Report LA-UR-15-23755, Los Alamos National Laboratory, 2015. To appear in J. Comput. Phys. (2015).

K. Lipnikov, J.D. Moulton and D. Svyatskiy, A Multilevel Multiscale Mimetic (M $^{3}$ ) method for two-phase flows in porous media. J. Comp. Phys. 227 (2008) 6727–6753. | DOI | MR | Zbl

K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115–152. | DOI | MR | Zbl

G. Manzini, A. Russo and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods. Math. Models Methods Appl. Sci. 24 (2014) 1665–1699. | DOI | MR | Zbl

L. Margolin, M. Shashkov and P. Smolarkiewicz, A discrete operator calculus for finite difference approximations. Comput. Methods Appl. Mech. Engrg. 187 (2000) 365–383. | DOI | MR | Zbl

A. Palha, P.P. Rebelo, R. Hiemstra, J. Kreeft and M. Gerritsma, Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys. B 257 (2014) 1394–1422. | DOI | MR | Zbl

N. Sukumar and E. Malsch, Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Engrg. 13 (2006) 129–163. | DOI | MR | Zbl

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Meth. Engrg. 61 (2004) 2045–2066. | DOI | MR | Zbl

E. Wachspress, A rational Finite Element Basis. Academic Press (1975). | MR | Zbl

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

J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83 (2014) 2101–2126. | DOI | MR | Zbl

Cité par Sources :