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Title: An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations

Abstract

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Speci cally, we investigate a Q2-Q1 mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the Q2-Q1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the nest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.

Authors:
 [1];  [2]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
  2. Sandia National Lab. (SNL-CA), Livermore, CA (United States). Center for Computing Research
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1399874
Alternate Identifier(s):
OSTI ID: 1394531
Report Number(s):
SAND-2016-6518J
Journal ID: ISSN 1099-1506; 644837; TRN: US1703220
Grant/Contract Number:  
AC04-94AL85000; AC05-00OR22725
Resource Type:
Accepted Manuscript
Journal Name:
Numerical Linear Algebra with Applications (Online)
Additional Journal Information:
Journal Name: Numerical Linear Algebra with Applications (Online); Journal Volume: 24; Journal Issue: 6; Journal ID: ISSN 1099-1506
Publisher:
Wiley
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; algebraic multigrid; mixed finite element discretizations; Navier–Stokes equations; preconditioning

Citation Formats

Prokopenko, Andrey, and Tuminaro, Raymond S. An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations. United States: N. p., 2016. Web. doi:10.1002/nla.2109.
Prokopenko, Andrey, & Tuminaro, Raymond S. An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations. United States. doi:10.1002/nla.2109.
Prokopenko, Andrey, and Tuminaro, Raymond S. Fri . "An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations". United States. doi:10.1002/nla.2109. https://www.osti.gov/servlets/purl/1399874.
@article{osti_1399874,
title = {An algebraic multigrid method for Q2-Q1 mixed discretizations of the Navier-Stokes equations},
author = {Prokopenko, Andrey and Tuminaro, Raymond S.},
abstractNote = {Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Speci cally, we investigate a Q2-Q1 mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the Q2-Q1 discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the nest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.},
doi = {10.1002/nla.2109},
journal = {Numerical Linear Algebra with Applications (Online)},
number = 6,
volume = 24,
place = {United States},
year = {2016},
month = {7}
}

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