High order VEM on curved domains

We deal with the virtual element method (VEM) for solving the Poisson equation on a domain $\Omega$ with curved boundaries. Given a polygonal approximation $\Omega_h$ of the domain $\Omega$, the standard order $m$ VEM [6], for $m$ increasing, leads to a suboptimal convergence rate. We adapt the approach of [14] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of $\Omega_h$, which, to retain computability, is evaluated after applying the projector $\Pi^\nabla$ onto the space of polynomials. Numerical experiments confirm the theory.

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

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

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

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

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

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

[7]  L. Beirao da Veiga,et al.  The Virtual Element Method with curved edges , 2017, ESAIM: Mathematical Modelling and Numerical Analysis.

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

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

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

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

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

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

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

[15]  Vidar Thomée,et al.  Polygonal Domain Approximation in Dirichlet's Problem , 1973 .

[16]  Lourenço Beirão da Veiga,et al.  Virtual element methods for parabolic problems on polygonal meshes , 2015 .

[17]  Antonio Huerta,et al.  Comparison of high‐order curved finite elements , 2011 .

[18]  J. Nitsche Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

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

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

[21]  Daniele A. Di Pietro,et al.  Assessment of Hybrid High-Order methods on curved meshes and comparison with discontinuous Galerkin methods , 2018, J. Comput. Phys..

[22]  K. Lipnikov On shape-regularity of polyhedral meshes for solving PDEs , 2013 .

[23]  Vidar Thomée,et al.  Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections , 1972 .

[24]  Todd F. Dupont L2 Error Estimates for Projection Methods for Parabolic Equations in Approximating Domains , 1974 .

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

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