Mimetic finite difference method for the stokes problem on polygonal meshes
Abstract
Various approaches to extend the finite element methods to nontraditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with loworder finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and nonconvex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the secondorder convergence for the velocity variable and the firstorder for the pressure.
 Authors:

 Los Alamos National Laboratory
 DIPARTIMENTO DI MATE
 PENNSYLVANIA STATE UNIV
 ISTIUTO DI MATEMATICA
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 956391
 Report Number(s):
 LAUR0900753; LAUR09753
Journal ID: ISSN 00219991; JCTPAH; TRN: US201013%%94
 DOE Contract Number:
 AC5206NA25396
 Resource Type:
 Journal Article
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Name: Journal of Computational Physics; Journal ID: ISSN 00219991
 Country of Publication:
 United States
 Language:
 English
 Subject:
 99; CONVERGENCE; ELASTICITY; FINITE DIFFERENCE METHOD; FINITE ELEMENT METHOD; FLEXIBILITY; GEOMETRY; IMPLEMENTATION; MATHEMATICAL MODELS; MATRICES; VELOCITY
Citation Formats
Lipnikov, K, Beirao Da Veiga, L, Gyrya, V, and Manzini, G. Mimetic finite difference method for the stokes problem on polygonal meshes. United States: N. p., 2009.
Web.
Lipnikov, K, Beirao Da Veiga, L, Gyrya, V, & Manzini, G. Mimetic finite difference method for the stokes problem on polygonal meshes. United States.
Lipnikov, K, Beirao Da Veiga, L, Gyrya, V, and Manzini, G. 2009.
"Mimetic finite difference method for the stokes problem on polygonal meshes". United States. https://www.osti.gov/servlets/purl/956391.
@article{osti_956391,
title = {Mimetic finite difference method for the stokes problem on polygonal meshes},
author = {Lipnikov, K and Beirao Da Veiga, L and Gyrya, V and Manzini, G},
abstractNote = {Various approaches to extend the finite element methods to nontraditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with loworder finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and nonconvex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the secondorder convergence for the velocity variable and the firstorder for the pressure.},
doi = {},
url = {https://www.osti.gov/biblio/956391},
journal = {Journal of Computational Physics},
issn = {00219991},
number = ,
volume = ,
place = {United States},
year = {2009},
month = {1}
}