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Title: SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations

Abstract

We present here the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov–Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection–diffusion–reaction problems in the convective-dominated regime. According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients. The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.

Authors:
 [1];  [1];  [2]
+ Show Author Affiliations
  1. Polytechnic Univ. of Turin (Italy). Dept. of Mathematics
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Inst. for Applied Mathematics and Information Technologies (IMATI), Pavia (Italy)
Publication Date:
Research Org.:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Polytechnic Univ. of Turin (Italy)
Sponsoring Org.:
USDOE Office of Science (SC), Fusion Energy Sciences (FES); USDOE National Nuclear Security Administration (NNSA); LANL Laboratory Directed Research and Development (LDRD) Program; Ministry of Education, Universities and Research (MIUR) (Italy)
OSTI Identifier:
1511223
Report Number(s):
LA-UR-17-20866
Journal ID: ISSN 0045-7825
Grant/Contract Number:  
AC52-06NA25396
Resource Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 340; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; virtual element methods; advection–diffusion–reaction problem; SUPG; stability; convergence

Citation Formats

Berrone, S., Borio, A., and Manzini, G.. SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations. United States: N. p., 2018. Web. doi:10.1016/j.cma.2018.05.027.
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Berrone, S., Borio, A., & Manzini, G.. SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations. United States. https://doi.org/10.1016/j.cma.2018.05.027
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Berrone, S., Borio, A., and Manzini, G.. Mon . "SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations". United States. https://doi.org/10.1016/j.cma.2018.05.027. https://www.osti.gov/servlets/purl/1511223.
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@article{osti_1511223,
title = {SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations},
author = {Berrone, S. and Borio, A. and Manzini, G.},
abstractNote = {We present here the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov–Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection–diffusion–reaction problems in the convective-dominated regime. According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients. The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.},
doi = {10.1016/j.cma.2018.05.027},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = ,
volume = 340,
place = {United States},
year = {2018},
month = {6}
}
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https://doi.org/10.1016/j.cma.2018.05.027
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Cited by: 28 works
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Figures / Tables:

Figure 1 Figure 1: Degrees of freedom of a hexagonal cell for $k$ = 1, 2, 3, 4; edge moments are marked by a circle; cell moments are marked by a square.
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Works referenced in this record:

Post processing of solution and flux for the nodal mimetic finite difference method: Post Processing of Nodal MFD Method
journal, July 2014

  • Beirão da Veiga, Lourenço; Manzini, Gianmarco; Putti, Mario
  • Numerical Methods for Partial Differential Equations, Vol. 31, Issue 1
  • DOI: 10.1002/num.21907

Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes
journal, January 2011

  • da Veiga, L. Beirão; Lipnikov, K.; Manzini, G.
  • SIAM Journal on Numerical Analysis, Vol. 49, Issue 5
  • DOI: 10.1137/100807764

Mimetic finite differences for elliptic problems
journal, December 2008

  • Brezzi, Franco; Buffa, Annalisa; Lipnikov, Konstantin
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 43, Issue 2
  • DOI: 10.1051/m2an:2008046

Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients
journal, January 2017

  • Manzini, G.; Lipnikov, K.; Moulton, J. D.
  • SIAM Journal on Numerical Analysis, Vol. 55, Issue 6
  • DOI: 10.1137/16M1108479

Mimetic finite difference method
journal, January 2014

  • Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
  • Journal of Computational Physics, Vol. 257
  • DOI: 10.1016/j.jcp.2013.07.031

A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes
journal, January 2016

  • Antonietti, P. F.; da Veiga, L. Beira͂o; Scacchi, S.
  • SIAM Journal on Numerical Analysis, Vol. 54, Issue 1
  • DOI: 10.1137/15M1008117

Basic principles of hp virtual elements on quasiuniform meshes
journal, June 2016

  • Beir ao da Veiga, L.; Chernov, A.; Mascotto, L.
  • Mathematical Models and Methods in Applied Sciences, Vol. 26, Issue 08
  • DOI: 10.1142/S021820251650038X

Virtual Element Method for general second-order elliptic problems on polygonal meshes
journal, February 2016

  • Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.
  • Mathematical Models and Methods in Applied Sciences, Vol. 26, Issue 04
  • DOI: 10.1142/S0218202516500160

Mixed virtual element methods for general second order elliptic problems on polygonal meshes
journal, May 2016

  • Beirão da Veiga, Lourenço; Brezzi, Franco; Marini, Luisa Donatella
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
  • DOI: 10.1051/m2an/2015067

Serendipity Nodal VEM spaces
journal, December 2016

A hybrid mortar virtual element method for discrete fracture network simulations
journal, February 2016

  • Benedetto, Matías Fernando; Berrone, Stefano; Borio, Andrea
  • Journal of Computational Physics, Vol. 306
  • DOI: 10.1016/j.jcp.2015.11.034

A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method
journal, February 2016

  • Benedetto, Matías Fernando; Berrone, Stefano; Scialò, Stefano
  • Finite Elements in Analysis and Design, Vol. 109
  • DOI: 10.1016/j.finel.2015.10.003

Towards effective flow simulations in realistic discrete fracture networks
journal, April 2016

  • Berrone, Stefano; Pieraccini, Sandra; Scialò, Stefano
  • Journal of Computational Physics, Vol. 310
  • DOI: 10.1016/j.jcp.2016.01.009

A virtual element method for contact
journal, September 2016

Residual a posteriori error estimation for the Virtual Element Method for elliptic problems
journal, March 2015

  • Beirão da Veiga, L.; Manzini, G.
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 2
  • DOI: 10.1051/m2an/2014047

A Parallel Solver for Large Scale DFN Flow Simulations
journal, January 2015

  • Berrone, Stefano; Pieraccini, Sandra; Scialò, Stefano
  • SIAM Journal on Scientific Computing, Vol. 37, Issue 3
  • DOI: 10.1137/140984014

Virtual element methods for parabolic problems on polygonal meshes: Vem For Parabolic Problems
journal, July 2015

  • Vacca, Giuseppe; Beirão da Veiga, Lourenco
  • Numerical Methods for Partial Differential Equations, Vol. 31, Issue 6
  • DOI: 10.1002/num.21982

The Hitchhiker's Guide to the Virtual Element Method
journal, May 2014

  • Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.
  • Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 08
  • DOI: 10.1142/S021820251440003X

The virtual element method for discrete fracture network simulations
journal, October 2014

  • Benedetto, Matías Fernando; Berrone, Stefano; Pieraccini, Sandra
  • Computer Methods in Applied Mechanics and Engineering, Vol. 280
  • DOI: 10.1016/j.cma.2014.07.016

Virtual Element Methods for plate bending problems
journal, January 2013

  • Brezzi, Franco; Marini, L. Donatella
  • Computer Methods in Applied Mechanics and Engineering, Vol. 253
  • DOI: 10.1016/j.cma.2012.09.012

Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method
journal, July 2017

Conforming polygonal finite elements
journal, January 2004

  • Sukumar, N.; Tabarraei, A.
  • International Journal for Numerical Methods in Engineering, Vol. 61, Issue 12
  • DOI: 10.1002/nme.1141

Convection-adapted BEM-based FEM: Convection-adapted BEM-based FEM
journal, June 2016

  • Hofreither, Clemens; Langer, Ulrich; Weißer, Steffen
  • ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 96, Issue 12
  • DOI: 10.1002/zamm.201500042

Residual error estimate for BEM-based FEM on polygonal meshes
journal, March 2011

Finite volume schemes for diffusion equations: Introduction to and review of modern methods
journal, May 2014

A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods
journal, February 2010

  • Droniou, JÉRÔMe; Eymard, Robert; GallouËT, Thierry
  • Mathematical Models and Methods in Applied Sciences, Vol. 20, Issue 02
  • DOI: 10.1142/S0218202510004222

Hybrid high-order methods for variable-diffusion problems on general meshes
journal, January 2015

hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes
journal, May 2014

  • Cangiani, Andrea; Georgoulis, Emmanuil H.; Houston, Paul
  • Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 10
  • DOI: 10.1142/S0218202514500146

Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009

  • Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho
  • SIAM Journal on Numerical Analysis, Vol. 47, Issue 2
  • DOI: 10.1137/070706616

Gradient schemes: Generic tools for the numerical analysis of diffusion equations
journal, May 2016

  • Droniou, Jérome; Eymard, Robert; Herbin, Raphaèle
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
  • DOI: 10.1051/m2an/2015079

Gradient Schemes: a Generic Framework for the Discretisation of Linear, Nonlinear and Nonlocal Elliptic and Parabolic Equations
journal, September 2013

  • Droniou, Jerome; Eymard, Robert; Gallouet, Thierry
  • Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 13
  • DOI: 10.1142/S0218202513500358

New perspectives on polygonal and polyhedral finite element methods
journal, May 2014

  • Manzini, Gianmarco; Russo, Alessandro; Sukumar, N.
  • Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 08
  • DOI: 10.1142/S0218202514400065

Hourglass stabilization and the virtual element method: Hourglass stabilization and the virtual element method
journal, January 2015

  • Cangiani, A.; Manzini, G.; Russo, A.
  • International Journal for Numerical Methods in Engineering, Vol. 102, Issue 3-4
  • DOI: 10.1002/nme.4854

Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes
journal, February 2018

  • Di Pietro, Daniele A.; Droniou, Jérôme; Manzini, Gianmarco
  • Journal of Computational Physics, Vol. 355
  • DOI: 10.1016/j.jcp.2017.11.018

Equivalent projectors for virtual element methods
journal, September 2013

The nonconforming virtual element method
journal, May 2016

  • Ayuso de Dios, Blanca; Lipnikov, Konstantin; Manzini, Gianmarco
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
  • DOI: 10.1051/m2an/2015090

A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation
journal, September 2014

The NonConforming Virtual Element Method for the Stokes Equations
journal, January 2016

  • Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco
  • SIAM Journal on Numerical Analysis, Vol. 54, Issue 6
  • DOI: 10.1137/15M1049531

The nonconforming virtual element method for plate bending problems
journal, July 2016

  • Zhao, Jikun; Chen, Shaochun; Zhang, Bei
  • Mathematical Models and Methods in Applied Sciences, Vol. 26, Issue 09
  • DOI: 10.1142/S021820251650041X

A non-conforming piecewise quadratic finite element on triangles
journal, April 1983

  • Fortin, M.; Soulie, M.
  • International Journal for Numerical Methods in Engineering, Vol. 19, Issue 4
  • DOI: 10.1002/nme.1620190405

Nonconforming finite elements for the Stokes problem
journal, May 1989

Crouzeix-Velte decompositions for higher-order finite elements
journal, March 2006

Gauss-Legendre elements: a stable, higher order non-conforming finite element family
journal, February 2007

Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation
journal, January 2005

Simple nonconforming quadrilateral Stokes element
journal, March 1992

  • Rannacher, R.; Turek, S.
  • Numerical Methods for Partial Differential Equations, Vol. 8, Issue 2
  • DOI: 10.1002/num.1690080202

Inf-sup stable non-conforming finite elements of arbitrary order on triangles
journal, November 2005

Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra
journal, September 2007

  • Matthies, Gunar
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 41, Issue 5
  • DOI: 10.1051/m2an:2007034

Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems
journal, December 2008

Further considerations on residual-free bubbles for advective-diffusive equations
journal, November 1998

  • Brezzi, F.; Franca, L. P.; Russo, A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 166, Issue 1-2
  • DOI: 10.1016/S0045-7825(98)00080-2

Stability of the residual free bubble method for bilinear finite elements on rectangular grids
journal, January 2002

Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
journal, September 1982

  • Brooks, Alexander N.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3
  • DOI: 10.1016/0045-7825(82)90071-8

Stabilized finite element methods: I. Application to the advective-diffusive model
journal, March 1992

  • Franca, Leopoldo P.; Frey, Sergio L.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 95, Issue 2
  • DOI: 10.1016/0045-7825(92)90143-8

Stabilized finite element schemes with LBB-stable elements for incompressible flows
journal, May 2005

  • Gelhard, Tobias; Lube, Gert; Olshanskii, Maxim A.
  • Journal of Computational and Applied Mathematics, Vol. 177, Issue 2
  • DOI: 10.1016/j.cam.2004.09.017

Analysis of the Space Semi-Discretized supg Method for Transient Convection–Diffusion Equations
journal, October 2011

  • Burman, Erik; Smith, Giles
  • Mathematical Models and Methods in Applied Sciences, Vol. 21, Issue 10
  • DOI: 10.1142/S0218202511005659

Consistent SUPG-method for transient transport problems: Stability and convergence
journal, March 2010

Order preserving SUPG stabilization for the virtual element formulation of advection–diffusion problems
journal, November 2016

  • Benedetto, M. F.; Berrone, S.; Borio, A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 311
  • DOI: 10.1016/j.cma.2016.07.043

A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method
journal, February 2016

  • Benedetto, Matías Fernando; Berrone, Stefano; Scialò, Stefano
  • Finite Elements in Analysis and Design, Vol. 109
  • DOI: 10.1016/j.finel.2015.10.003

A hybrid mortar virtual element method for discrete fracture network simulations
journal, February 2016

  • Benedetto, Matías Fernando; Berrone, Stefano; Borio, Andrea
  • Journal of Computational Physics, Vol. 306
  • DOI: 10.1016/j.jcp.2015.11.034

Towards effective flow simulations in realistic discrete fracture networks
journal, April 2016

  • Berrone, Stefano; Pieraccini, Sandra; Scialò, Stefano
  • Journal of Computational Physics, Vol. 310
  • DOI: 10.1016/j.jcp.2016.01.009

A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs
journal, January 2016

  • Berrone, Stefano; Borio, Andrea; Scialò, Stefano
  • SIAM Journal on Numerical Analysis, Vol. 54, Issue 1
  • DOI: 10.1137/15m1014760

A posteriori error estimates for the virtual element method
preprint, January 2016

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    Works referencing / citing this record:

    SUPG-stabilized virtual elements for diffusion-convection problems: a robustness analysis
    journal, September 2021

    • Beirão da Veiga, Lourenco; Dassi, Franco; Lovadina, Carlo
    • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 55, Issue 5
    • DOI: 10.1051/m2an/2021050
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