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

Gyrya, Vitaliy; Lipnikov, Konstantin; Manzini, Gianmarco

doi:10.1051/m2an/2015088

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

Journal Article · Sun May 01 00:00:00 EDT 2016 · Mathematical Modelling and Numerical Analysis

DOI:https://doi.org/10.1051/m2an/2015088· OSTI ID:1304825

Gyrya, Vitaliy ^[1]; Lipnikov, Konstantin ^[1]; Manzini, Gianmarco ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
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)

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.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1304825

Report Number(s):: LA-UR-15-22806

Journal Information:: Mathematical Modelling and Numerical Analysis, Journal Name: Mathematical Modelling and Numerical Analysis Journal Issue: 3 Vol. 50; ISSN 0764-583X

Country of Publication:: United States

Language:: English

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A high-order discontinuous Galerkin approach to the elasto-acoustic problem Antonietti, Paola F.; Bonaldi, Francesco; Mazzieri, Ilario arXiv https://doi.org/10.48550/arxiv.1803.01351	preprint	January 2018
The virtual element method for resistive magnetohydrodynamics Alvarez, S. Naranjo; Bokil, V. A.; Gyrya, V. arXiv https://doi.org/10.48550/arxiv.2004.11467	preprint	January 2020
The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems Manzini, Gianmarco; Maguolo, Gianluca; Putti, Mario Journal of Scientific Computing, Vol. 80, Issue 3 https://doi.org/10.1007/s10915-019-01002-4	journal	July 2019
The fully nonconforming virtual element method for biharmonic problems Antonietti, P. F.; Manzini, G.; Verani, M. Mathematical Models and Methods in Applied Sciences, Vol. 28, Issue 02 https://doi.org/10.1142/s0218202518500100	journal	December 2017

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97 MATHEMATICS AND COMPUTING
Mathematics
Mimetic finite difference method
polygonal mesh
polyhedral mesh
high-order discretization
diffusion equation in mixed form
Poisson problem

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

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