The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review ismotivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.

[1]  J. Waals.  The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density , 1979 .

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

[3]  Charles M. Elliott,et al.  A second order splitting method for the Cahn-Hilliard equation , 1989 .

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

[5]  高等学校計算数学学報編輯委員会編,et al.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[6]  Thirupathi Gudi,et al.  An interior penalty method for a sixth-order elliptic equation , 2011 .

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

[8]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[9]  C. M. Elliott,et al.  Numerical Studies of the Cahn-Hilliard Equation for Phase Separation , 1987 .

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

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

[12]  C. Lovadina,et al.  A virtual element method for the von Kármán equations , 2021, ESAIM: Mathematical Modelling and Numerical Analysis.

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

[14]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[15]  C. M. Elliott,et al.  A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation , 1989 .

[16]  Filippo Gazzola,et al.  Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains , 2010 .

[17]  David Mora,et al.  Virtual element for the buckling problem of Kirchhoff–Love plates , 2020 .

[18]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

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

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

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

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

[23]  F. Gazzola,et al.  Polyharmonic Boundary Value Problems , 2010 .

[24]  D. Komatitsch,et al.  Introduction to the spectral element method for three-dimensional seismic wave propagation , 1999 .

[25]  Thomas J. R. Hughes,et al.  A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework , 2014 .

[26]  Mira Schedensack,et al.  A New Discretization for mth-Laplace Equations with Arbitrary Polynomial Degrees , 2015, SIAM J. Numer. Anal..

[27]  L. Donatella Marini,et al.  Virtual Element Method for fourth order problems: L2-estimates , 2016, Comput. Math. Appl..

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

[29]  Ezio Faccioli,et al.  Spectral-domain decomposition methods for the solution of acoustic and elastic wave equations , 1996 .

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

[31]  Heng Chi,et al.  On nonconvex meshes for elastodynamics using virtual element methods with explicit time integration , 2019, Computer Methods in Applied Mechanics and Engineering.

[32]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[33]  David Mora,et al.  A virtual element method for the vibration problem of Kirchhoff plates , 2017, ESAIM: Mathematical Modelling and Numerical Analysis.

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

[35]  P. Antonietti,et al.  The virtual element method for linear elastodynamics models. Design, analysis, and implementation , 2019, ArXiv.

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

[37]  J. S. Rowlinson,et al.  Translation of J. D. van der Waals' “The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density” , 1979 .

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

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

[40]  Felipe Lepe,et al.  A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges , 2015, Journal of Scientific Computing.

[41]  Richard S. Falk,et al.  A mixed-Lagrange multiplier finite element method for the polyharmonic equation , 1985 .

[42]  Charles M. Elliott,et al.  Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation , 1992 .

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

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

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

[46]  David Mora,et al.  A virtual element method for the transmission eigenvalue problem , 2018, Mathematical Models and Methods in Applied Sciences.

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

[48]  G. Azzone,et al.  Bridge closure in the road network of Lombardy: a spatio-temporal analysis of the socio-economic impacts , 2022, Statistical Methods & Applications.

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

[50]  Simone Vantini,et al.  The Importance of Being a Band: Finite-Sample Exact Distribution-Free Prediction Sets for Functional Data , 2021 .

[51]  Shaochun Chen,et al.  The Morley-Type Virtual Element for Plate Bending Problems , 2018, J. Sci. Comput..

[52]  Lorenzo Mascotto,et al.  Exponential convergence of the hp virtual element method in presence of corner singularities , 2017, Numerische Mathematik.

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

[54]  Paola F. Antonietti,et al.  A high-order discontinuous Galerkin approach to the elasto-acoustic problem , 2018, Computer Methods in Applied Mechanics and Engineering.

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

[56]  Stefano Berrone,et al.  A Parallel Solver for Large Scale DFN Flow Simulations , 2015, SIAM J. Sci. Comput..

[57]  Peter Wriggers,et al.  Phase-field modeling of brittle fracture using an efficient virtual element scheme , 2018, Computer Methods in Applied Mechanics and Engineering.

[58]  Dietmar Gallistl,et al.  Stable splitting of polyharmonic operators by generalized Stokes systems , 2017, Math. Comput..

[59]  L. Mascotto,et al.  Exploring high-order three dimensional virtual elements: Bases and stabilizations , 2017, Comput. Math. Appl..

[60]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[61]  Paola F. Antonietti,et al.  High-order Discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes , 2018, Computer Methods in Applied Mechanics and Engineering.

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

[63]  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.

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

[65]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

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

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

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

[69]  John W. Barrett,et al.  Finite element approximation of a sixth order nonlinear degenerate parabolic equation , 2004, Numerische Mathematik.

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

[71]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[73]  Long Chen,et al.  Nonconforming Virtual Element Method for 2mth Order Partial Differential Equations in ℝn , 2018, Math. Comput..

[74]  Franco Dassi,et al.  A C1 Virtual Element Method on polyhedral meshes , 2018, Comput. Math. Appl..

[75]  Monique Dauge,et al.  Polynomials in the Sobolev World , 2007 .

[76]  Krishna Garikipati,et al.  A discontinuous Galerkin method for the Cahn-Hilliard equation , 2006, J. Comput. Phys..

[77]  Axel Voigt,et al.  A new phase-field model for strongly anisotropic systems , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[79]  Endre Süli,et al.  Discontinuous Galerkin Finite Element Approximation of the Cahn-Hilliard Equation with Convection , 2009, SIAM J. Numer. Anal..

[80]  Ming Wang,et al.  Minimal finite element spaces for 2m-th-order partial differential equations in Rn , 2012, Math. Comput..

[81]  Stefano Berrone,et al.  Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method , 2017 .

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

[83]  Xin Liu,et al.  A virtual element method for the Cahn-Hilliard problem in mixed form , 2019, Appl. Math. Lett..

[84]  Giuseppe Vacca,et al.  Virtual Element Methods for hyperbolic problems on polygonal meshes , 2016, Comput. Math. Appl..

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

[86]  Franck Pigeonneau,et al.  A Hybrid High-Order Method for the Cahn-Hilliard problem in Mixed Form , 2015, SIAM J. Numer. Anal..

[87]  Blanca Ayuso de Dios,et al.  Stability Analysis of Discontinuous Galerkin Approximations to the Elastodynamics Problem , 2013, J. Sci. Comput..

[88]  Kyoungsoo Park,et al.  Numerical recipes for elastodynamic virtual element methods with explicit time integration , 2019, International Journal for Numerical Methods in Engineering.

[89]  K. Cheng.  Theory of Superconductivity , 1948, Nature.

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

[91]  E. Favvas,et al.  What is spinodal decomposition , 2008 .

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

[93]  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..

[94]  Stig Larsson,et al.  THE CAHN-HILLIARD EQUATION , 2007 .

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

[96]  A. Quarteroni,et al.  Non-conforming high order approximations of the elastodynamics equation , 2012 .

[97]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[98]  Feng Chen,et al.  Efcient Energy Stable Schemes with Spectral Discretization in Space for Anisotropic Cahn-Hilliard Systems , 2020 .