An algebraic multigrid method for Q2Q1 mixed discretizations of the NavierStokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Speci cally, we investigate a Q _{2}Q _{1} mixed finite element discretization of the incompressible NavierStokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degreesoffreedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever aging this colocated structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the Q _{2}Q _{1} 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 intergrid transfers, an energy minimization AMG (EMINAMG) is utilized. EMINAMG 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 NavierStokes problems.
 Authors:

^{[1]};
^{[2]}
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States). Center for Computing Research
 Sandia National Lab. (SNLCA), Livermore, CA (United States). Center for Computing Research
 Publication Date:
 Report Number(s):
 SAND20166518J
Journal ID: ISSN 10991506; 644837; TRN: US1703220
 Grant/Contract Number:
 AC0494AL85000
 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 10991506
 Publisher:
 Wiley
 Research Org:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
 OSTI Identifier:
 1399874
Prokopenko, Andrey, and Tuminaro, Raymond S. An algebraic multigrid method for Q2Q1 mixed discretizations of the NavierStokes equations. United States: N. p.,
Web. doi:10.1002/nla.2109.
Prokopenko, Andrey, & Tuminaro, Raymond S. An algebraic multigrid method for Q2Q1 mixed discretizations of the NavierStokes equations. United States. doi:10.1002/nla.2109.
Prokopenko, Andrey, and Tuminaro, Raymond S. 2016.
"An algebraic multigrid method for Q2Q1 mixed discretizations of the NavierStokes equations". United States.
doi:10.1002/nla.2109. https://www.osti.gov/servlets/purl/1399874.
@article{osti_1399874,
title = {An algebraic multigrid method for Q2Q1 mixed discretizations of the NavierStokes 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 colocated at mesh points. Speci cally, we investigate a Q2Q1 mixed finite element discretization of the incompressible NavierStokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degreesoffreedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches lever aging this colocated structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the Q2Q1 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 intergrid transfers, an energy minimization AMG (EMINAMG) is utilized. EMINAMG 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 NavierStokes problems.},
doi = {10.1002/nla.2109},
journal = {Numerical Linear Algebra with Applications (Online)},
number = 6,
volume = 24,
place = {United States},
year = {2016},
month = {7}
}