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

Journal Article · · Computer Methods in Applied Mechanics and Engineering
 [1];  [1];  [2]
  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)

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.

Research Organization:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Polytechnic Univ. of Turin (Italy)
Sponsoring Organization:
USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24); USDOE National Nuclear Security Administration (NNSA); LANL Laboratory Directed Research and Development (LDRD) Program; Ministry of Education, Universities and Research (MIUR) (Italy)
Grant/Contract Number:
AC52-06NA25396
OSTI ID:
1511223
Report Number(s):
LA-UR--17-20866; {"","Journal ID: ISSN 0045-7825"}
Journal Information:
Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 340; ISSN 0045-7825
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (63)

Consistent SUPG-method for transient transport problems: Stability and convergence journal March 2010
Equivalent projectors for virtual element methods journal September 2013
The NonConforming Virtual Element Method for the Stokes Equations journal January 2016
Stabilized finite element methods: I. Application to the advective-diffusive model journal March 1992
Convection-adapted BEM-based FEM: Convection-adapted BEM-based FEM
  • Hofreither, Clemens; Langer, Ulrich; Weißer, Steffen
  • ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 96, Issue 12 https://doi.org/10.1002/zamm.201500042
journal June 2016
A Parallel Solver for Large Scale DFN Flow Simulations journal January 2015
Stability of the residual free bubble method for bilinear finite elements on rectangular grids journal January 2002
A virtual element method for contact journal September 2016
Basic principles of hp virtual elements on quasiuniform meshes journal June 2016
The nonconforming virtual element method journal May 2016
Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra journal September 2007
Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method journal July 2017
Residual error estimate for BEM-based FEM on polygonal meshes journal March 2011
Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes journal February 2018
Post processing of solution and flux for the nodal mimetic finite difference method: Post Processing of Nodal MFD Method
  • Beirão da Veiga, Lourenço; Manzini, Gianmarco; Putti, Mario
  • Numerical Methods for Partial Differential Equations, Vol. 31, Issue 1 https://doi.org/10.1002/num.21907
journal July 2014
New perspectives on polygonal and polyhedral finite element methods journal May 2014
Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients journal January 2017
Finite volume schemes for diffusion equations: Introduction to and review of modern methods journal May 2014
A hybrid mortar virtual element method for discrete fracture network simulations journal February 2016
A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes journal January 2016
Residual a posteriori error estimation for the Virtual Element Method for elliptic problems journal March 2015
Gauss-Legendre elements: a stable, higher order non-conforming finite element family journal February 2007
A posteriori error estimates for the virtual element method preprint January 2016
Crouzeix-Velte decompositions for higher-order finite elements journal March 2006
The Hitchhiker's Guide to the Virtual Element Method journal May 2014
Virtual Element Method for general second-order elliptic problems on polygonal meshes journal February 2016
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations journal September 1982
Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods: Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods
  • Natarajan, Sundararajan; Bordas, Stéphane PA; Ooi, Ean Tat
  • International Journal for Numerical Methods in Engineering, Vol. 104, Issue 13 https://doi.org/10.1002/nme.4965
journal July 2015
Mixed virtual element methods for general second order elliptic problems on polygonal meshes journal May 2016
Towards effective flow simulations in realistic discrete fracture networks journal April 2016
Virtual element methods for parabolic problems on polygonal meshes: Vem For Parabolic Problems journal July 2015
Further considerations on residual-free bubbles for advective-diffusive equations journal November 1998
A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs journal January 2016
Conforming polygonal finite elements journal January 2004
A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods journal February 2010
A non-conforming piecewise quadratic finite element on triangles journal April 1983
Hybrid high-order methods for variable-diffusion problems on general meshes journal January 2015
A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method journal February 2016
Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems journal December 2008
Stabilized finite element schemes with LBB-stable elements for incompressible flows journal May 2005
Gradient schemes: Generic tools for the numerical analysis of diffusion equations journal May 2016
Mimetic finite differences for elliptic problems journal December 2008
Inf-sup stable non-conforming finite elements of arbitrary order on triangles journal November 2005
Gradient Schemes: a Generic Framework for the Discretisation of Linear, Nonlinear and Nonlocal Elliptic and Parabolic Equations journal September 2013
A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs journal January 2016
Nonconforming finite elements for the Stokes problem journal May 1989
Virtual Element Methods for plate bending problems journal January 2013
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems journal January 2009
Hourglass stabilization and the virtual element method: Hourglass stabilization and the virtual element method journal January 2015
Analysis of the Space Semi-Discretized supg Method for Transient Convection–Diffusion Equations journal October 2011
Towards effective flow simulations in realistic discrete fracture networks journal April 2016
Simple nonconforming quadrilateral Stokes element journal March 1992
Mimetic finite difference method journal January 2014
Serendipity Nodal VEM spaces journal December 2016
A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation journal September 2014
hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes journal May 2014
The virtual element method for discrete fracture network simulations journal October 2014
Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation journal January 2005
A hybrid mortar virtual element method for discrete fracture network simulations journal February 2016
Order preserving SUPG stabilization for the virtual element formulation of advection–diffusion problems journal November 2016
Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes journal January 2011
The nonconforming virtual element method for plate bending problems journal July 2016
A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method journal February 2016

Cited By (1)

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