Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method

In this paper we propose a modified construction for the polynomial basis on polygons used in the Virtual Element Method (VEM). This construction is alternative to the usual monomial basis used in the classical construction of the VEM and is designed in order to improve numerical stability. For badly shaped elements the construction of the projection matrices required for assembling the local coefficients of the linear system within the VEM discretization of Partial Differential Equations can result very ill conditioned. The proposed approach can be easily implemented within an existing VEM code in order to reduce the possible ill conditioning of the elemental projection matrices. Numerical results applied to an hydro-geological flow simulation that often produces very badly shaped elements show a clear improvement of the quality of the numerical solution, confirming the viability of the approach. The method can be conveniently combined with a classical implementation of the VEM and applied element-wise, thus requiring a rather moderate additional numerical cost.

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

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

[3]  C. Fidelibus,et al.  A code for scaled flow simulations on generated fracture networks , 1999 .

[4]  C. Fidelibus,et al.  Derivation of equivalent pipe network analogues for three‐dimensional discrete fracture networks by the boundary element method , 1999 .

[5]  J. Erhel,et al.  A Mortar BDD Method for Solving Flow in Stochastic Discrete Fracture Networks , 2014 .

[6]  Stefano Berrone,et al.  A PDE-Constrained Optimization Formulation for Discrete Fracture Network Flows , 2013, SIAM J. Sci. Comput..

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

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

[9]  Jean-Raynald de Dreuzy,et al.  Synthetic benchmark for modeling flow in 3D fractured media , 2013, Comput. Geosci..

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

[11]  Benoit Noetinger,et al.  A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks , 2012, J. Comput. Phys..

[12]  G. Marsily,et al.  Modeling fracture flow with a stochastic discrete fracture network: calibration and validation: 1. The flow model , 1990 .

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

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

[15]  Raducanu Razvan,et al.  MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES , 2012 .

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

[17]  Stefano Berrone,et al.  The Virtual Element Method for Underground Flow Simulations in Fractured Media , 2016 .

[18]  Stefano Berrone,et al.  Non-stationary transport phenomena in networks of fractures: Effective simulations and stochastic analysis , 2017 .

[19]  Stefano Berrone,et al.  An optimization approach for large scale simulations of discrete fracture network flows , 2014, J. Comput. Phys..

[20]  Stefano Berrone,et al.  A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs , 2016, SIAM J. Numer. Anal..

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

[22]  Sandra Pieraccini,et al.  On a PDE-Constrained Optimization Approach for Flow Simulations in Fractured Media , 2016 .

[23]  Benoit Noetinger,et al.  A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks accounting for matrix to fracture flow , 2015, J. Comput. Phys..

[24]  C. Fidelibus,et al.  The 2D hydro‐mechanically coupled response of a rock mass with fractures via a mixed BEM–FEM technique , 2007 .

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

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

[27]  Stefano Berrone,et al.  Towards effective flow simulations in realistic discrete fracture networks , 2016, J. Comput. Phys..

[28]  Claudio Canuto,et al.  Uncertainty quantification in Discrete Fracture Network models: Stochastic fracture transmissivity , 2015, Comput. Math. Appl..

[29]  Jeffrey D. Hyman,et al.  Conforming Delaunay Triangulation of Stochastically Generated Three Dimensional Discrete Fracture Networks: A Feature Rejection Algorithm for Meshing Strategy , 2014, SIAM J. Sci. Comput..

[30]  Stefano Berrone,et al.  On Simulations of Discrete Fracture Network Flows with an Optimization-Based Extended Finite Element Method , 2013, SIAM J. Sci. Comput..

[31]  J. Jaffré,et al.  Modeling flow in porous media with fractures; Discrete fracture models with matrix-fracture exchange , 2012 .

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

[33]  C. Fidelibus,et al.  A BEM solution of steady-state flow problems in discrete fracture networks with minimization of core storage , 2003 .

[34]  Pierre M. Adler,et al.  Fractures and Fracture Networks , 1999 .

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

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

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

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

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

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

[41]  Jean-Raynald de Dreuzy,et al.  A Generalized Mixed Hybrid Mortar Method for Solving Flow in Stochastic Discrete Fracture Networks , 2012, SIAM J. Sci. Comput..

[42]  J. Erhel,et al.  A mixed hybrid Mortar method for solving flow in discrete fracture networks , 2010 .