Mimetic finite difference method for the stokes problem on polygonal meshes

Lipnikov, K; Beirao Da Veiga, L; Gyrya, V; Manzini, G

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

Full Record
Other Related Research

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:

Lipnikov, K ^[1]; Beirao Da Veiga, L ^[2]; Gyrya, V ^[3]; Manzini, G ^[4]

Los Alamos National Laboratory
DIPARTIMENTO DI MATE
PENNSYLVANIA STATE UNIV
ISTIUTO DI MATEMATICA

Publication Date:: 2009-01-01

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.

Copy to clipboard


                    Lipnikov, K, Beirao Da Veiga, L, Gyrya, V, & Manzini, G. Mimetic finite difference method for the stokes problem on polygonal meshes.  United States.

Copy to clipboard


                    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.

Copy to clipboard


                    
@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          = {},

  url          = {https://www.osti.gov/biblio/956391},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = ,

  volume       = ,

  place        = {United States},

  year         = {2009},

  month        = {1}

}

Copy to clipboard

Journal Article:

View Journal Article

Other availability

Find in Google Scholar

Search WorldCat to find libraries that may hold this journal

Save / Share:

Export Metadata

Save to My Library

Similar records in OSTI.GOV collections:

A mimetic finite difference method for the Stokes problem with elected edge bubbles

Conference Lipnikov, K; Berirao, L

A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The unstable P{sub 1}-P{sub 0} discretization is stabilized by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments. The discretizations schemes for Stokes and Navier-Stokes equations must satisfy the celebrated inf-sup (or the LBB) stability condition. The stability condition implies a balance between discrete spaces for velocity and pressure. In finite elements, this balance is frequently achieved by adding bubble functions to the velocity space. The goal of this articlemore »« less
Full Text Available
Arbitrary Order Mixed Mimetic Finite Differences Method with Nodal Degrees of Freedom

Technical Report Iaroshenko, Oleksandr; Gyrya, Vitaliy; Manzini, Gianmarco

In this work we consider a modification to an arbitrary order mixed mimetic finite difference method (MFD) for a diffusion equation on general polygonal meshes [1]. The modification is based on moving some degrees of freedom (DoF) for a flux variable from edges to vertices. We showed that for a non-degenerate element this transformation is locally equivalent, i.e. there is a one-to-one map between the new and the old DoF. Globally, on the other hand, this transformation leads to a reduction of the total number of degrees of freedom (by up to 40%) and additional continuity of the discrete flux.
https://doi.org/10.2172/1321697

Full Text Available
A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

Journal Article Manzini, Gianmarco; Mazzia, Annamaria - Journal of Computational Dynamics

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problemmore »« less
https://doi.org/10.3934/jcd.2021020

Full Text Available
A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

Technical Report Manzini, Gianmarco; Mazzia, Annamaria

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method (FEM) for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problemmore »« less
https://doi.org/10.2172/1787266

Full Text Available
The arbitrary order mimetic finite difference method for a diffusion equation with a non-symmetric diffusion tensor

Journal Article Gyrya, V.; Lipnikov, K. - Journal of Computational Physics

Here, we present the arbitrary order mimetic finite difference (MFD) discretization for the diffusion equation with non-symmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes. The diffusion tensor is assumed to be positive definite. The asymmetry of the diffusion tensor requires changes to the standard MFD construction. We also present new approach for the construction that guarantees positive definiteness of the non-symmetric mass matrix in the space of discrete velocities. The numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme. For higher orders schemes, wemore »« less
Cited by 1
https://doi.org/10.1016/j.jcp.2017.07.019

Full Text Available

Similar Records