Annotations on the virtual element method for second-order elliptic problems

This document contains working annotations on the Virtual Element Method (VEM) for the approximate solution of diffusion problems with variable coefficients. To read this document you are assumed to have familiarity with concepts from the numerical discretization of Partial Differential Equations (PDEs) and, in particular, the Finite Element Method (FEM). This document is not an introduction to the FEM, for which many textbooks are available. Eventually, this document will evolve into a tutorial introduction to the VEM (but this is really a long-term goal).

[1]  Raytcho D. Lazarov,et al.  Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems , 2009, SIAM J. Numer. Anal..

[2]  Jerome Droniou,et al.  FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS: INTRODUCTION TO AND REVIEW OF MODERN METHODS , 2014, 1407.1567.

[3]  M. Putti,et al.  Post processing of solution and flux for the nodal mimetic finite difference method , 2015 .

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

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

[6]  R. Eymard,et al.  3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids , 2008 .

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

[8]  Gianmarco Manzini,et al.  The Discrete Duality Finite Volume Method for Convection-diffusion Problems , 2010, SIAM J. Numer. Anal..

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

[10]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[11]  Lourenço Beirão da Veiga,et al.  A mimetic discretization of elliptic obstacle problems , 2013, Math. Comput..

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

[13]  Robert Eymard,et al.  A mixed finite volume scheme for anisotropic diffusion problems on any grid , 2006, Numerische Mathematik.

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

[15]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

[16]  Thierry Gallouët,et al.  GRADIENT SCHEMES: A GENERIC FRAMEWORK FOR THE DISCRETISATION OF LINEAR, NONLINEAR AND NONLOCAL ELLIPTIC AND PARABOLIC EQUATIONS , 2013 .

[17]  Bernardo Cockburn,et al.  The hybridizable discontinuous Galerkin methods , 2011 .

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

[19]  Junping Wang,et al.  A weak Galerkin finite element method for second-order elliptic problems , 2011, J. Comput. Appl. Math..

[20]  Gian Luca Delzanno,et al.  New approach for the study of linear Vlasov stability of inhomogeneous systems , 2006 .

[21]  S. Mohammadi Extended Finite Element Method , 2008 .

[22]  Gianmarco Manzini,et al.  The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient , 2016, J. Comput. Phys..

[23]  Gianmarco Manzini,et al.  Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes , 2011, SIAM J. Numer. Anal..

[24]  N. Sukumar,et al.  Extended finite element method on polygonal and quadtree meshes , 2008 .

[25]  T. Rabczuk,et al.  On three-dimensional modelling of crack growth using partition of unity methods , 2010 .

[26]  Gianmarco Manzini,et al.  ARBITRARY ORDER NODAL MIMETIC DISCRETIZATIONS OF ELLIPTIC PROBLEMS ON POLYGONAL MESHES WITH ARBITRARY REGULAR SOLUTION , 2012 .

[27]  Anna Scotti,et al.  MIMETIC FINITE DIFFERENCE APPROXIMATION OF FLOWS IN FRACTURED POROUS MEDIA , 2016 .

[28]  Alexandre Ern,et al.  Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes , 2012, 1211.3354.

[29]  L. Beirao da Veiga,et al.  A Virtual Element Method for elastic and inelastic problems on polytope meshes , 2015, 1503.02042.

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

[31]  Alessio Fumagalli,et al.  Dual Virtual Element Method for Discrete Fractures Networks , 2016, SIAM J. Sci. Comput..

[32]  Gianmarco Manzini,et al.  Discretization of Mixed Formulations of Elliptic Problems on Polyhedral Meshes , 2016, 1610.05850.

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

[34]  L. Beirao da Veiga,et al.  Serendipity Nodal VEM spaces , 2015, 1510.08477.

[35]  R. Eymard,et al.  A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS , 2008, 0812.2097.

[36]  Gianmarco Manzini,et al.  M-Adaptation in the mimetic finite difference method , 2014 .

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

[38]  Haiying Wang,et al.  Superconvergent discontinuous Galerkin methods for second-order elliptic problems , 2009, Math. Comput..

[39]  Marc I. Gerritsma,et al.  A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations , 2016, J. Comput. Phys..

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

[41]  Simone Scacchi,et al.  A C1 Virtual Element Method for the Cahn-Hilliard Equation with Polygonal Meshes , 2015, SIAM J. Numer. Anal..

[42]  Markus H. Gross,et al.  Polyhedral Finite Elements Using Harmonic Basis Functions , 2008, Comput. Graph. Forum.

[43]  Gian Luca Delzanno,et al.  A Legendre-Fourier spectral method with exact conservation laws for the Vlasov-Poisson system , 2016, J. Comput. Phys..

[44]  Franco Brezzi,et al.  The Hitchhiker's Guide to the Virtual Element Method , 2014 .

[45]  Gianmarco Manzini,et al.  A Higher-Order Formulation of the Mimetic Finite Difference Method , 2008, SIAM J. Sci. Comput..

[46]  Gianmarco Manzini,et al.  Convergence analysis of the high-order mimetic finite difference method , 2009, Numerische Mathematik.

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

[48]  Junping Wang,et al.  A computational study of the weak Galerkin method for second-order elliptic equations , 2011, Numerical Algorithms.

[49]  I. Babuska,et al.  Acta Numerica 2003: Survey of meshless and generalized finite element methods: A unified approach , 2003 .

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

[51]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[52]  Glaucio H. Paulino,et al.  On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes , 2014 .

[53]  T. Belytschko,et al.  The extended/generalized finite element method: An overview of the method and its applications , 2010 .

[54]  Paola F. Antonietti,et al.  Mimetic finite differences for nonlinear and control problems , 2014 .

[55]  Arun L. Gain Polytope-based topology optimization using a mimetic-inspired method , 2014 .

[56]  Chandrajit L. Bajaj,et al.  Interpolation error estimates for mean value coordinates over convex polygons , 2011, Adv. Comput. Math..

[57]  Ivo Babuška,et al.  Generalized finite element methods for three-dimensional structural mechanics problems , 2000 .

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

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

[60]  E. Wachspress,et al.  A Rational Finite Element Basis , 1975 .

[61]  F. Hermeline Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes , 2007 .

[62]  A. Ern,et al.  A Hybrid High-Order method for the incompressible Navier-Stokes equations based on Temam's device , 2018, J. Comput. Phys..

[63]  Gianmarco Manzini,et al.  Monotonicity Conditions in the Mimetic Finite Difference Method , 2011 .

[64]  L. Beirao da Veiga,et al.  Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes , 2014 .

[65]  Kai Hormann,et al.  A general construction of barycentric coordinates over convex polygons , 2006, Adv. Comput. Math..

[66]  Gabriel N. Gatica,et al.  A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem , 2017 .

[67]  Francesca Gardini,et al.  Virtual element method for second-order elliptic eigenvalue problems , 2016, 1610.03675.

[68]  Glaucio H. Paulino,et al.  Polygonal finite elements for topology optimization: A unifying paradigm , 2010 .

[69]  L. Beirao da Veiga,et al.  Basic principles of hp virtual elements on quasiuniform meshes , 2015, 1508.02242.

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

[71]  Joseph E. Bishop,et al.  A displacement‐based finite element formulation for general polyhedra using harmonic shape functions , 2014 .

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

[73]  Alexandre Ern,et al.  An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators , 2014, Comput. Methods Appl. Math..

[74]  Alexandre Ern,et al.  Hybrid high-order methods for variable-diffusion problems on general meshes , 2015 .

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

[76]  I. Babuska,et al.  GENERALIZED FINITE ELEMENT METHODS — MAIN IDEAS, RESULTS AND PERSPECTIVE , 2004 .

[77]  Gianmarco Manzini,et al.  Convergence of the mimetic finite difference method for eigenvalue problems in mixed form , 2011 .

[78]  Eugenio Oñate,et al.  The meshless finite element method , 2003 .

[79]  T. Belytschko,et al.  Extended finite element method for three-dimensional crack modelling , 2000 .

[80]  Gianmarco Manzini,et al.  Mimetic scalar products of discrete differential forms , 2014, J. Comput. Phys..

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

[82]  Joe D. Warren,et al.  Barycentric coordinates for convex polytopes , 1996, Adv. Comput. Math..

[83]  Pascal Omnes,et al.  A FINITE VOLUME METHOD FOR THE LAPLACE EQUATION ON ALMOST ARBITRARY TWO-DIMENSIONAL GRIDS , 2005 .

[84]  F. Hermeline,et al.  A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes , 2000 .

[85]  Gian Luca Delzanno,et al.  Multi-dimensional, fully-implicit, spectral method for the Vlasov-Maxwell equations with exact conservation laws in discrete form , 2015, J. Comput. Phys..

[86]  YE JUNPINGWANGANDXIU A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS , 2014 .

[87]  Gianmarco Manzini,et al.  Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients , 2016, SIAM J. Numer. Anal..

[88]  Shan Zhao,et al.  WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS. , 2013, Journal of computational physics.

[89]  Michael S. Floater,et al.  Gradient Bounds for Wachspress Coordinates on Polytopes , 2013, SIAM J. Numer. Anal..

[90]  Lourenço Beirão da Veiga,et al.  Hierarchical A Posteriori Error Estimators for the Mimetic Discretization of Elliptic Problems , 2013, SIAM J. Numer. Anal..

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

[92]  Gianmarco Manzini,et al.  Recent techniques for PDE discretizations on polyhedral meshes , 2014 .

[93]  A. Ern,et al.  Mathematical Aspects of Discontinuous Galerkin Methods , 2011 .

[94]  Gianmarco Manzini,et al.  The arbitrary order mixed mimetic finite difference method for the diffusion equation , 2016 .

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

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

[97]  Paola F. Antonietti,et al.  Mimetic Discretizations of Elliptic Control Problems , 2013, J. Sci. Comput..

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

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

[100]  Annalisa Buffa,et al.  Mimetic finite differences for elliptic problems , 2009 .

[101]  Glaucio H. Paulino,et al.  Some basic formulations of the virtual element method (VEM) for finite deformations , 2017 .

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

[103]  Michael Ortiz,et al.  Local Maximum-Entropy Approximation Schemes , 2007 .

[104]  Gianmarco Manzini,et al.  The Discrete Duality Finite Volume Method for Stokes Equations on Three-Dimensional Polyhedral Meshes , 2012, SIAM J. Numer. Anal..

[105]  Gianmarco Manzini,et al.  Flux reconstruction and solution post-processing in mimetic finite difference methods , 2008 .

[106]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

[107]  Paola F. Antonietti,et al.  Mimetic finite difference approximation of quasilinear elliptic problems , 2015 .

[108]  Eugene L. Wachspress,et al.  Rational bases for convex polyhedra , 2010, Comput. Math. Appl..

[109]  F. Hermeline,et al.  Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes , 2003 .

[110]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .

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

[112]  Pavel B. Bochev,et al.  Principles of Mimetic Discretizations of Differential Operators , 2006 .

[113]  Alexandre Ern,et al.  Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods , 2016 .

[114]  Glaucio H. Paulino,et al.  Topology optimization using polytopes , 2013, 1312.7016.

[115]  Oliver J. Sutton,et al.  The virtual element method in 50 lines of MATLAB , 2016, Numerical Algorithms.

[116]  David Mora,et al.  A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem , 2016, Comput. Math. Appl..

[117]  Stefano Berrone,et al.  Order preserving SUPG stabilization for the Virtual Element formulation of advection-diffusion problems , 2016 .

[118]  Gian Luca Delzanno,et al.  On the velocity space discretization for the Vlasov-Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods , 2016, Comput. Phys. Commun..

[119]  Gianmarco Manzini,et al.  A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation , 2014, J. Comput. Phys..

[120]  Erwin Laure,et al.  Spectral Solver for Multi-scale Plasma Physics Simulations with Dynamically Adaptive Number of Moments , 2015, ICCS.

[121]  Gianmarco Manzini,et al.  A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems , 2011 .

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

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

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

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

[126]  Gianmarco Manzini,et al.  An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems , 2008 .

[127]  Gianmarco Manzini,et al.  Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems , 2011, J. Comput. Phys..

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

[129]  Odd Andersen,et al.  On the use of the Virtual Element Method for geomechanics on reservoir grids , 2016 .