The arbitrary order mixed mimetic finite difference method for the diffusion equation
Abstract
Here, we propose an arbitraryorder 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 wellposedness of the resulting approximation. The method also requires the definition of a highorder 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 highorder 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.
 Authors:

 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Consiglio Nazionale delle Ricerche (IMATICNR), Pavia (Italy); Centro di Simulazione Numerica Avanzata (CeSNA)  IUSS Pavia, Pavia (Italy)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1304825
 Report Number(s):
 LAUR1522806
Journal ID: ISSN 0764583X
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Mathematical Modelling and Numerical Analysis
 Additional Journal Information:
 Journal Volume: 50; Journal Issue: 3; Journal ID: ISSN 0764583X
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Mathematics; Mimetic finite difference method, polygonal mesh, polyhedral mesh, highorder discretization, diffusion equation in mixed form, Poisson problem
Citation Formats
Gyrya, Vitaliy, Lipnikov, Konstantin, and Manzini, Gianmarco. The arbitrary order mixed mimetic finite difference method for the diffusion equation. United States: N. p., 2016.
Web. doi:10.1051/m2an/2015088.
Gyrya, Vitaliy, Lipnikov, Konstantin, & Manzini, Gianmarco. The arbitrary order mixed mimetic finite difference method for the diffusion equation. United States. doi:10.1051/m2an/2015088.
Gyrya, Vitaliy, Lipnikov, Konstantin, and Manzini, Gianmarco. Sun .
"The arbitrary order mixed mimetic finite difference method for the diffusion equation". United States. doi:10.1051/m2an/2015088. https://www.osti.gov/servlets/purl/1304825.
@article{osti_1304825,
title = {The arbitrary order mixed mimetic finite difference method for the diffusion equation},
author = {Gyrya, Vitaliy and Lipnikov, Konstantin and Manzini, Gianmarco},
abstractNote = {Here, we propose an arbitraryorder 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 wellposedness of the resulting approximation. The method also requires the definition of a highorder 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 highorder 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},
journal = {Mathematical Modelling and Numerical Analysis},
number = 3,
volume = 50,
place = {United States},
year = {2016},
month = {5}
}
Web of Science