Exponential convergence of the hp virtual element method in presence of corner singularities

In the present work, we analyze the hp version of virtual element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the hp finite element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in h and $$p$$p is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between hp virtual elements and hp finite elements.

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

[2]  Konstantin Lipnikov,et al.  Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes , 2005, SIAM J. Numer. Anal..

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

[4]  P. F. Antonietti,et al.  A multigrid algorithm for the $p$-version of the Virtual Element Method , 2017, 1703.02285.

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

[6]  Ralf Hiptmair,et al.  Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-DGFEM , 2012 .

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

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

[9]  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..

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

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

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

[13]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[14]  P. Houston,et al.  hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes , 2017 .

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

[16]  Emmanuil H. Georgoulis,et al.  Inverse-type estimates on hp-finite element spaces and applications , 2008, Math. Comput..

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

[18]  Chandrajit L. Bajaj,et al.  Error estimates for generalized barycentric interpolation , 2010, Adv. Comput. Math..

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

[20]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[21]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[22]  L. Tartar.  An Introduction to Sobolev Spaces and Interpolation Spaces , 2007 .

[23]  Ivo Babuška,et al.  The optimal convergence rate of the p-version of the finite element method , 1987 .

[24]  Jens Markus Melenk,et al.  HP-INTERPOLATION OF NON-SMOOTH FUNCTIONS , 2003 .

[25]  Lorenzo Mascotto,et al.  The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains , 2017, IMA Journal of Numerical Analysis.

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

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

[28]  Jens Markus Melenk,et al.  hp-Interpolation of Nonsmooth Functions and an Application to hp-A posteriori Error Estimation , 2005, SIAM J. Numer. Anal..

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

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

[31]  Lorenzo Mascotto,et al.  Exponential convergence of the hp Virtual Element Method with corner singularities , 2016, 1611.10165.

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

[33]  Lourenco Beirao da Veiga,et al.  Stability Analysis for the Virtual Element Method , 2016, 1607.05988.

[34]  Jie Shen,et al.  Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle , 2010, Math. Comput..

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

[36]  Ivo Babuška,et al.  Regularity of the solution of elliptic problems with piecewise analytic data, II: the trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions , 1989 .

[37]  Ivonne Sgura,et al.  Virtual Element Method for the Laplace-Beltrami equation on surfaces , 2016, 1612.02369.

[38]  Moshe Dubiner.  Spectral methods on triangles and other domains , 1991 .

[39]  B. Guo,et al.  REGULARITY OF THE SOLUTION OF ELLIPTIC PROBLEMS WITH PIECEWISE ANALYTIC DATA , Ih THE TRACE SPACES AND APPLICATION TO THE BOUNDARY VALUE PROBLEMS WITH NONHOMOGENEOUS BOUNDARY CONDITIONS , 2022 .

[40]  Ivo Babuška,et al.  Regularity of the solution of elliptic problems with piecewise analytic data. Part 1. Boundary value problems for linear elliptic equation of second order , 1988 .

[41]  P. Tesini,et al.  On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations , 2012, J. Comput. Phys..

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

[43]  Gianmarco Manzini,et al.  The Mimetic Finite Difference Method for Elliptic Problems , 2014 .

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

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

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

[47]  Sergej Rjasanow,et al.  Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Higher Order Bem-based Fem on Polygonal Meshes Higher Order Bem-based Fem on Polygonal Meshes Higher Order Bem-based Fem on Polygonal Meshes , 2022 .

[48]  T. Koornwinder.  Two-Variable Analogues of the Classical Orthogonal Polynomials , 1975 .

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

[50]  Lourenço Beirão da Veiga,et al.  A Stream Virtual Element Formulation of the Stokes Problem on Polygonal Meshes , 2014, SIAM J. Numer. Anal..

[51]  Ivo Babuška,et al.  The h-p version of the finite element method , 1986 .

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

[53]  Yvon Maday,et al.  Polynomial interpolation results in Sobolev spaces , 1992 .