Conforming virtual element approximations of the two-dimensional Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes. The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the L and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.

[1]  L. Beirao da Veiga,et al.  Divergence free Virtual Elements for the Stokes problem on polygonal meshes , 2015, 1510.01655.

[2]  Felipe Lepe,et al.  A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges , 2021, J. Sci. Comput..

[3]  Konstantin Lipnikov,et al.  A Mimetic Discretization of the Stokes Problem with Selected Edge Bubbles , 2010, SIAM J. Sci. Comput..

[4]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[5]  Gianmarco Manzini,et al.  A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem , 2021, Journal of Computational Dynamics.

[6]  Emmanuil H. Georgoulis,et al.  A posteriori error estimates for the virtual element method , 2016, Numerische Mathematik.

[7]  L. Beirao da Veiga,et al.  The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows , 2018, Journal of Scientific Computing.

[8]  Eugene Wachspress.  Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics , 2015 .

[9]  Stefano Berrone,et al.  Towards effective flow simulations in realistic discrete fracture networks , 2016, J. Comput. Phys..

[10]  G. Manzini,et al.  The virtual element method for resistive magnetohydrodynamics , 2020, Computer Methods in Applied Mechanics and Engineering.

[11]  Peter Wriggers,et al.  A virtual element method for contact , 2016 .

[12]  Brian J. Kirby,et al.  Microfluidic transport in microdevices for rare cell capture , 2012, Electrophoresis.

[13]  L. Beirao da Veiga,et al.  H(div) and H(curl)-conforming VEM , 2014, 1407.6822.

[14]  Ahmed Alsaedi,et al.  Equivalent projectors for virtual element methods , 2013, Comput. Math. Appl..

[15]  G. Manzini,et al.  Virtual elements for Maxwell's equations , 2021, Comput. Math. Appl..

[16]  Shaochun Chen,et al.  The nonconforming virtual element method for plate bending problems , 2016 .

[17]  Gianmarco Manzini,et al.  The NonConforming Virtual Element Method for the Stokes Equations , 2016, SIAM J. Numer. Anal..

[18]  Gianmarco Manzini,et al.  Conforming and nonconforming virtual element methods for elliptic problems , 2015, 1507.03543.

[19]  G. Vacca,et al.  Sharper error estimates for Virtual Elements and a bubble-enriched version , 2020, SIAM J. Numer. Anal..

[20]  Gianmarco Manzini,et al.  The nonconforming Virtual Element Method for eigenvalue problems , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[21]  G. Vacca,et al.  The p- and hp-versions of the virtual element method for elliptic eigenvalue problems , 2018, Comput. Math. Appl..

[22]  J. M. Hyman,et al.  Mimetic discretizations for Maxwell equations and the equations of magnetic diffusion , 1998 .

[23]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[24]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[25]  Franco Dassi,et al.  Bricks for the mixed high-order virtual element method: Projectors and differential operators , 2018, Applied Numerical Mathematics.

[26]  Ilaria Perugia,et al.  A Plane Wave Virtual Element Method for the Helmholtz Problem , 2015, 1505.04965.

[27]  P. F. Antonietti,et al.  The fully nonconforming virtual element method for biharmonic problems , 2016, 1611.08736.

[28]  Stéphane Bordas,et al.  Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods , 2015 .

[29]  Glaucio H. Paulino,et al.  Bridging art and engineering using Escher-based virtual elements , 2015 .

[30]  Gianmarco Manzini,et al.  A posteriori error estimation and adaptivity in hp virtual elements , 2018, Numerische Mathematik.

[31]  Mikhail Shashkov,et al.  Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes , 2004 .

[32]  Stefano Berrone,et al.  The virtual element method for discrete fracture network simulations , 2014 .

[33]  Stefano Berrone,et al.  A hybrid mortar virtual element method for discrete fracture network simulations , 2016, J. Comput. Phys..

[34]  Gianmarco Manzini,et al.  Hourglass stabilization and the virtual element method , 2015 .

[35]  L. B. D. Veiga,et al.  A virtual element method with arbitrary regularity , 2014 .

[36]  Giuseppe Vacca,et al.  Virtual Elements for the Navier-Stokes Problem on Polygonal Meshes , 2017, SIAM J. Numer. Anal..

[37]  Lorenzo Mascotto,et al.  Ill‐conditioning in the virtual element method: Stabilizations and bases , 2017, 1705.10581.

[38]  Panayot S. Vassilevski,et al.  Mixed finite element methods for incompressible flow: Stationary Stokes equations , 2010 .

[39]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[40]  Børre Bang,et al.  Application of homogenization theory related to Stokes flow in porous media , 1999 .

[41]  A. Russo,et al.  New perspectives on polygonal and polyhedral finite element methods , 2014 .

[42]  Gianmarco Manzini,et al.  Convergence Analysis of the Mimetic Finite Difference Method for Elliptic Problems , 2009, SIAM J. Numer. Anal..

[43]  Andreas Wiegmann,et al.  Specialized methods for direct numerical simulations in porous media , 2019 .

[44]  G. Manzini,et al.  Extended virtual element method for the Laplace problem with singularities and discontinuities , 2019, Computer Methods in Applied Mechanics and Engineering.

[45]  F. Dassi,et al.  Parallel solvers for virtual element discretizations of elliptic equations in mixed form , 2019, Comput. Math. Appl..

[46]  Alessandro Russo,et al.  Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes , 2014, 1506.07328.

[47]  Gianmarco Manzini,et al.  Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes , 2017, J. Comput. Phys..

[48]  G. Manzini,et al.  SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations , 2018, Computer Methods in Applied Mechanics and Engineering.

[49]  N. Sukumar,et al.  Conforming polygonal finite elements , 2004 .

[50]  Richard Hofer.  Sedimentation of Inertialess Particles in Stokes Flows , 2016, 1610.03748.

[51]  Richard S. Falk,et al.  Basic principles of mixed Virtual Element Methods , 2014 .

[52]  Gianmarco Manzini,et al.  Residual a posteriori error estimation for the Virtual Element Method for elliptic problems , 2015 .

[53]  Gianmarco Manzini,et al.  The Virtual Element Method for Eigenvalue Problems with Potential Terms on Polytopic Meshes , 2018, Applications of Mathematics.

[54]  Stefano Berrone,et al.  A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs , 2016, SIAM J. Numer. Anal..

[55]  Gianmarco Manzini,et al.  Mimetic finite difference method for the Stokes problem on polygonal meshes , 2009, J. Comput. Phys..

[56]  Giovanni P. Galdi,et al.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems , 2011 .

[57]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[58]  K. Lipnikov,et al.  The nonconforming virtual element method , 2014, 1405.3741.

[59]  M. Shashkov,et al.  A new discretization methodology for diffusion problems on generalized polyhedral meshes , 2007 .

[60]  L. Beirao da Veiga,et al.  The Stokes complex for Virtual Elements in three dimensions , 2019, Mathematical Models and Methods in Applied Sciences.

[61]  M. Shashkov,et al.  CONVERGENCE OF MIMETIC FINITE DIFFERENCE METHOD FOR DIFFUSION PROBLEMS ON POLYHEDRAL MESHES WITH CURVED FACES , 2006 .

[62]  Y. Kuznetsov,et al.  New mixed finite element method on polygonal and polyhedral meshes , 2005 .

[63]  Lorenzo Mascotto,et al.  p- and hp- virtual elements for the Stokes problem , 2020, Advances in Computational Mathematics.

[64]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[65]  Franco Brezzi,et al.  Virtual Element Methods for plate bending problems , 2013 .

[66]  Franco Dassi,et al.  The Mixed Virtual Element Method on curved edges in two dimensions , 2020, ArXiv.

[67]  T. Dupont,et al.  Polynomial approximation of functions in Sobolev spaces , 1980 .

[68]  Gianmarco Manzini,et al.  The role of mesh quality and mesh quality indicators in the virtual element method , 2021, Advances in Computational Mathematics.

[69]  Gianmarco Manzini,et al.  Virtual Element Methods for Elliptic Problems on Polygonal Meshes , 2017 .

[70]  Franco Dassi,et al.  The mixed virtual element method for grids with curved interfaces , 2020, ArXiv.

[71]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[72]  Gianmarco Manzini,et al.  Mimetic finite difference method , 2014, J. Comput. Phys..

[73]  G. Burton.  Sobolev Spaces , 2013 .

[74]  Paola F. Antonietti,et al.  The conforming virtual element method for polyharmonic problems , 2018, Comput. Math. Appl..

[75]  Lourenço Beirão da Veiga,et al.  Virtual Elements for Linear Elasticity Problems , 2013, SIAM J. Numer. Anal..

[76]  Gianmarco Manzini,et al.  Error Analysis for a Mimetic Discretization of the Steady Stokes Problem on Polyhedral Meshes , 2010, SIAM J. Numer. Anal..

[77]  Stefano Berrone,et al.  A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method , 2016 .