###
An algebraic multigrid method for *Q* _{2}- *Q* _{1} mixed discretizations of the Navier-Stokes equations: AMG for *Q* _{2}- *Q* _{1} discretization

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. Specifically, we investigate a Q

_{2}-Q_{1}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 leveraging this colocated structure are not applicable. This article 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 finest grid. To define coefficients within the intergrid 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.- Publication Date:

- Report Number(s):
- SAND-2016-6518J

Journal ID: ISSN 1070-5325; 103531

- Grant/Contract Number:
- AC05-00OR22725; AC04-94AL85000

- Type:
- Accepted Manuscript

- Journal Name:
- Numerical Linear Algebra with Applications

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 6; Journal ID: ISSN 1070-5325

- Publisher:
- Wiley

- Research Org:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA (United States)

- Sponsoring Org:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; algebraic multigrid; mixed finite element discretizations; Navier–Stokes equations; preconditioning

- OSTI Identifier:
- 1399874

- Alternate Identifier(s):
- OSTI ID: 1394531

```
Prokopenko, Andrey, and Tuminaro, Raymond S.
```*An algebraic multigrid method for Q 2-Q 1 mixed discretizations of the Navier-Stokes equations: AMG for Q 2-Q 1 discretization*. United States: N. p.,
Web. doi:10.1002/nla.2109.

```
Prokopenko, Andrey, & Tuminaro, Raymond S.
```*An algebraic multigrid method for Q 2-Q 1 mixed discretizations of the Navier-Stokes equations: AMG for Q 2-Q 1 discretization*. United States. doi:10.1002/nla.2109.

```
Prokopenko, Andrey, and Tuminaro, Raymond S. 2017.
"An algebraic multigrid method for Q 2-Q 1 mixed discretizations of the Navier-Stokes equations: AMG for Q 2-Q 1 discretization". United States.
doi:10.1002/nla.2109. https://www.osti.gov/servlets/purl/1399874.
```

```
@article{osti_1399874,
```

title = {An algebraic multigrid method for Q 2-Q 1 mixed discretizations of the Navier-Stokes equations: AMG for Q 2-Q 1 discretization},

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. Specifically, 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 leveraging this colocated structure are not applicable. This article 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 finest grid. To define coefficients within the intergrid 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},

number = 6,

volume = 24,

place = {United States},

year = {2017},

month = {6}

}