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

Abstract

Here, 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.

Authors:
 [1];  [1];  [2]
  1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Consiglio Nazionale delle Ricerche (IMATI-CNR), 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):
LA-UR-15-22806
Journal ID: ISSN 0764-583X
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Mathematical Modelling and Numerical Analysis
Additional Journal Information:
Journal Volume: 50; Journal Issue: 3; Journal ID: ISSN 0764-583X
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics; Mimetic finite difference method, polygonal mesh, polyhedral mesh, high-order 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 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},
journal = {Mathematical Modelling and Numerical Analysis},
issn = {0764-583X},
number = 3,
volume = 50,
place = {United States},
year = {2016},
month = {5}
}

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