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Title: Mimetic finite difference method for the stokes problem on polygonal meshes

Abstract

Various approaches to extend the finite element methods to non-traditional 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 low-order 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 non-convex 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 second-order convergence for the velocity variable and the first-order for the pressure.

Authors:
 [1];  [2];  [3];  [4]
  1. Los Alamos National Laboratory
  2. DIPARTIMENTO DI MATE
  3. PENNSYLVANIA STATE UNIV
  4. 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):
LA-UR-09-00753; LA-UR-09-753
Journal ID: ISSN 0021-9991; JCTPAH; TRN: US201013%%94
DOE Contract Number:  
AC52-06NA25396
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Name: Journal of Computational Physics; Journal ID: ISSN 0021-9991
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. Thu . "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 non-traditional 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 low-order 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 non-convex 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 second-order convergence for the velocity variable and the first-order for the pressure.},
doi = {},
journal = {Journal of Computational Physics},
issn = {0021-9991},
number = ,
volume = ,
place = {United States},
year = {2009},
month = {1}
}