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

Berrone, S.; Borio, A.; Manzini, G.

doi:10.1016/j.cma.2018.05.027

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:

Berrone, S. ^[1]; Borio, A. ^[1]; Manzini, G. ^[2]

Polytechnic Univ. of Turin (Italy). Dept. of Mathematics
Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Inst. for Applied Mathematics and Information Technologies (IMATI), Pavia (Italy)

Publication Date:: 2018-06-18

Research Org.:: Los Alamos National Lab. (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.  https://doi.org/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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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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