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.
DOI: 10.1051/m2an/2015088
Mots-clés : Mimetic finite difference method, polygonal mesh, high-order discretization, Poisson problem, mixed formulation
@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://www.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://www.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://www.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, Volume 50 (2016) no. 3, pp. 851-877. doi : 10.1051/m2an/2015088. http://www.numdam.org/articles/10.1051/m2an/2015088/
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