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Title: Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations.

Abstract

No abstract prepared.

Authors:
;  [1]; ; ;
  1. (School of Mathematics, University of Manchester, Manchester, UK)
Publication Date:
Research Org.:
Sandia National Laboratories
Sponsoring Org.:
USDOE
OSTI Identifier:
897133
Report Number(s):
SAND2006-2072J
TRN: US200704%%551
DOE Contract Number:
AC04-94AL85000
Resource Type:
Journal Article
Resource Relation:
Journal Name: Proposed for publication in the SIAM Journal on Scientific Computing.
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; NAVIER-STOKES EQUATIONS; LEAST SQUARE FIT; DISCRETE ORDINATE METHOD

Citation Formats

Shadid, John Nicolas, Silvester, David, Elman, Howard, Howle, Victoria E., and Tuminaro, Raymond Stephen. Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations.. United States: N. p., 2006. Web.
Shadid, John Nicolas, Silvester, David, Elman, Howard, Howle, Victoria E., & Tuminaro, Raymond Stephen. Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations.. United States.
Shadid, John Nicolas, Silvester, David, Elman, Howard, Howle, Victoria E., and Tuminaro, Raymond Stephen. Sat . "Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations.". United States. doi:.
@article{osti_897133,
title = {Least squares preconditioners for stabilized discretizations of the Navier-Stokes equations.},
author = {Shadid, John Nicolas and Silvester, David and Elman, Howard and Howle, Victoria E. and Tuminaro, Raymond Stephen},
abstractNote = {No abstract prepared.},
doi = {},
journal = {Proposed for publication in the SIAM Journal on Scientific Computing.},
number = ,
volume = ,
place = {United States},
year = {Sat Apr 01 00:00:00 EST 2006},
month = {Sat Apr 01 00:00:00 EST 2006}
}
  • 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 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 lever- aging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocitymore » 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 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.« less
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  • No abstract prepared.
  • Formulation of locally conservative least-squares finite element methods (LSFEMs) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require non-standard boundary conditions (Bochev and Gunzburger, 2009 [3]), while methods that admit the no-slip condition satisfy the incompressibility equation only approximately (Bochev and Gunzburger, 2009 [4, Chapter 7]). Here we address this problem by proving a new non-standard stability bound for the velocity–vorticity–pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximatedmore » by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. Here, we also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces (Kreeft et al., 2011) and report several numerical tests, which confirm its mimetic properties.« less