On mass-conserving least-squares methods

J. J. Heys, Eunjung Lee, T. A. Manteuffel, S. F. Mccormick

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.

Original languageEnglish
Pages (from-to)1675-1693
Number of pages19
JournalSIAM Journal on Scientific Computing
Volume28
Issue number5
DOIs
Publication statusPublished - 2006 Dec 1

Fingerprint

Mass Conservation
Least Square Method
Conservation
Stationary Navier-Stokes Equations
Boundary conditions
Algebraic multigrid
Discrete Operators
Navier Stokes equations
First-order System
Elliptic Partial Differential Equations
Reformulation
Variational Methods
Positive definite
Least Squares
Three-dimension
Branching
Tube
Navier-Stokes Equations
Approximate Solution
Bifurcation

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

Heys, J. J., Lee, E., Manteuffel, T. A., & Mccormick, S. F. (2006). On mass-conserving least-squares methods. SIAM Journal on Scientific Computing, 28(5), 1675-1693. https://doi.org/10.1137/050640928
Heys, J. J. ; Lee, Eunjung ; Manteuffel, T. A. ; Mccormick, S. F. / On mass-conserving least-squares methods. In: SIAM Journal on Scientific Computing. 2006 ; Vol. 28, No. 5. pp. 1675-1693.
@article{8a50512aadbe42dd85f32a83d86680ef,
title = "On mass-conserving least-squares methods",
abstract = "Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.",
author = "Heys, {J. J.} and Eunjung Lee and Manteuffel, {T. A.} and Mccormick, {S. F.}",
year = "2006",
month = "12",
day = "1",
doi = "10.1137/050640928",
language = "English",
volume = "28",
pages = "1675--1693",
journal = "SIAM Journal of Scientific Computing",
issn = "1064-8275",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "5",

}

Heys, JJ, Lee, E, Manteuffel, TA & Mccormick, SF 2006, 'On mass-conserving least-squares methods', SIAM Journal on Scientific Computing, vol. 28, no. 5, pp. 1675-1693. https://doi.org/10.1137/050640928

On mass-conserving least-squares methods. / Heys, J. J.; Lee, Eunjung; Manteuffel, T. A.; Mccormick, S. F.

In: SIAM Journal on Scientific Computing, Vol. 28, No. 5, 01.12.2006, p. 1675-1693.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On mass-conserving least-squares methods

AU - Heys, J. J.

AU - Lee, Eunjung

AU - Manteuffel, T. A.

AU - Mccormick, S. F.

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.

AB - Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.

UR - http://www.scopus.com/inward/record.url?scp=34548432013&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548432013&partnerID=8YFLogxK

U2 - 10.1137/050640928

DO - 10.1137/050640928

M3 - Article

VL - 28

SP - 1675

EP - 1693

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 5

ER -