A simple and effective gradient recovery scheme and a posteriori error estimator for the Virtual Element Method (VEM)

Abstract This paper introduces a general recovery-based a posteriori error estimation framework for the Virtual Element Method (VEM) of arbitrary order on general polygonal/polyhedral meshes. The framework consists of a gradient recovery scheme and a posteriori error estimator based on the recovered displacement gradient. A skeletal error, which accurately mimics the behavior of the L 2 error of the displacement gradient by only sampling the displacement gradient on the mesh skeleton, is introduced. Through numerical studies on various polygonal/polyhedral meshes, we demonstrate that the proposed gradient recovery scheme can produce considerably more accurate displacement gradient than the original VEM solutions, and that the a posteriori error estimator is able to accurately capture both local and global errors without the knowledge of exact solutions. We also demonstrate the use of the VEM skeletal error estimators to guide adaptive mesh refinement.

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

[2]  Franco Dassi,et al.  High-order Virtual Element Method on polyhedral meshes , 2017, Comput. Math. Appl..

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

[4]  Alessandro Colombo,et al.  Agglomeration-based physical frame dG discretizations: An attempt to be mesh free , 2014 .

[5]  Robert L. Taylor,et al.  VEM for Inelastic Solids , 2018 .

[6]  Ted Belytschko,et al.  Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements , 1994 .

[7]  Chandrajit L. Bajaj,et al.  Quadratic serendipity finite elements on polygons using generalized barycentric coordinates , 2011, Math. Comput..

[8]  Zhimin Zhang,et al.  A Posteriori Error Estimates Based on the Polynomial Preserving Recovery , 2004, SIAM J. Numer. Anal..

[9]  Glaucio H. Paulino,et al.  Polygonal multiresolution topology optimization (PolyMTOP) for structural dynamics , 2016 .

[10]  Rüdiger Verfürth,et al.  A Posteriori Error Estimation Techniques for Finite Element Methods , 2013 .

[11]  Jean B. Lasserre,et al.  Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra , 2015, Computational Mechanics.

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

[13]  Amir R. Khoei,et al.  A polygonal finite element method for modeling arbitrary interfaces in large deformation problems , 2012 .

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

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

[16]  E. Artioli,et al.  Arbitrary order 2D virtual elements for polygonal meshes: part II, inelastic problem , 2017, Computational Mechanics.

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

[18]  G. Paulino,et al.  PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab , 2012 .

[19]  H. Nguyen-Xuan.  A polytree‐based adaptive polygonal finite element method for topology optimization , 2017 .

[20]  Ahmed Naga,et al.  THE POLYNOMIAL-PRESERVING RECOVERY FOR HIGHER ORDER FINITE ELEMENT METHODS IN 2D AND 3D , 2005 .

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

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

[23]  Bijan Boroomand,et al.  RECOVERY BY EQUILIBRIUM IN PATCHES (REP) , 1997 .

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

[25]  M. Ainsworth,et al.  A posteriori error estimators in the finite element method , 1991 .

[26]  Peter Wriggers,et al.  A Virtual Element Method for 2D linear elastic fracture analysis , 2018, Computer Methods in Applied Mechanics and Engineering.

[27]  N. Sukumar.  Construction of polygonal interpolants: a maximum entropy approach , 2004 .

[28]  O. C. Zienkiewicz,et al.  The superconvergent patch recovery (SPR) and adaptive finite element refinement , 1992 .

[29]  Joseph E. Bishop,et al.  Simulating the pervasive fracture of materials and structures using randomly close packed Voronoi tessellations , 2009 .

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

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

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

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

[34]  Simona Perotto,et al.  On some new recovery-based a posteriori error estimators , 2006 .

[35]  Glaucio H. Paulino,et al.  Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture , 2012 .

[36]  Glaucio H. Paulino,et al.  Unstructured polygonal meshes with adaptive refinement for the numerical simulation of dynamic cohesive fracture , 2014, International Journal of Fracture.

[37]  Glaucio H. Paulino,et al.  Polygonal finite elements for incompressible fluid flow , 2014 .

[38]  Alessandro Russo,et al.  $$H({\text {div}})$$H(div) and $$H(\mathbf{curl})$$H(curl)-conforming virtual element methods , 2016 .

[39]  Peter Wriggers,et al.  Polygonal finite element methods for contact-impact problems on non-conformal meshes , 2014 .

[40]  Alvise Sommariva,et al.  Gauss-Green cubature and moment computation over arbitrary geometries , 2009, J. Comput. Appl. Math..

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

[42]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[43]  Lourenço Beirão da Veiga,et al.  H(div)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H({\text {div}})$$\end{document} and H(curl)\documentclass[12pt] , 2015, Numerische Mathematik.

[44]  Peter Wriggers,et al.  A low order 3D virtual element formulation for finite elasto–plastic deformations , 2017, Computational Mechanics.

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

[46]  Bijan Boroomand,et al.  An improved REP recovery and the effectivity robustness test , 1997 .

[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]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[49]  M. Floater.  Mean value coordinates , 2003, Computer Aided Geometric Design.

[50]  Steffen Weißer,et al.  Universität Des Saarlandes Fachrichtung 6.1 – Mathematik Residual Error Estimate for Bem-based Fem on Polygonal Meshes Residual Error Estimate for Bem-based Fem on Polygonal Meshes Residual Error Estimate for Bem-based Fem on Polygonal Meshes , 2022 .

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

[52]  Glaucio H. Paulino,et al.  PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes , 2012 .

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

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

[55]  Peter Wriggers,et al.  Efficient virtual element formulations for compressible and incompressible finite deformations , 2017 .

[56]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[57]  Zhimin Zhang.  POLYNOMIAL PRESERVING GRADIENT RECOVERY AND A POSTERIORI ESTIMATE FOR BILINEAR ELEMENT ON IRREGULAR QUADRILATERALS , 2004 .

[58]  P. Steinmann,et al.  Adaptive Poly-FEM for the analysis of plane elasticity problems , 2017 .

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

[60]  S. de Miranda,et al.  A stress/displacement Virtual Element method for plane elasticity problems , 2017, 1702.01702.

[61]  Leszek Demkowicz,et al.  High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) methods using ultraweak formulations , 2017, 1706.06754.

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

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

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

[65]  Glaucio H. Paulino,et al.  Polygonal finite elements for finite elasticity , 2015 .

[66]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .