Geodesic
The arbitrary order mixed mimetic finite difference method for the diffusion equation
Gyrya, Vitaliy1 ; Lipnikov, Konstantin1  ; Manzini, Gianmarco1, 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

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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 :
DOI : 10.1051/m2an/2015088
Classification : 65N30, 65N12, 65G99, 76R99
Keywords: Mimetic finite difference method, polygonal mesh, high-order discretization, Poisson problem, mixed formulation

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 
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     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},
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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

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