Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations

Hirofumi Notsu; Masahisa Tabata

doi:10.1007/978-4-431-56457-7_18

Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations

Hirofumi Notsu^*, Masahisa Tabata

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

4 Citations (Scopus)

Abstract

Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitkäranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

Original language	English
Title of host publication	Mathematical Fluid Dynamics, Present and Future
Publisher	Springer New York LLC
Pages	497-530
Number of pages	34
Volume	183
ISBN (Print)	9784431564553
DOIs	https://doi.org/10.1007/978-4-431-56457-7_18
Publication status	Published - 2016
Event	8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014 - Tokyo, Japan Duration: 2014 Nov 11 → 2014 Nov 14

Other

Other	8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014
Country/Territory	Japan
City	Tokyo
Period	14/11/11 → 14/11/14

Keywords

Error estimates
Navier-Strokes equations
Second-order scheme
Stabilized Lagrange-Galerkin scheme

ASJC Scopus subject areas

Mathematics(all)

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10.1007/978-4-431-56457-7_18

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Notsu, H & Tabata, M 2016, Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations. in Mathematical Fluid Dynamics, Present and Future. vol. 183, Springer New York LLC, pp. 497-530, 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014, Tokyo, Japan, 14/11/11. https://doi.org/10.1007/978-4-431-56457-7_18

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title = "Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations",

abstract = "Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitk{\"a}ranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.",

keywords = "Error estimates, Navier-Strokes equations, Second-order scheme, Stabilized Lagrange-Galerkin scheme",

author = "Hirofumi Notsu and Masahisa Tabata",

year = "2016",

doi = "10.1007/978-4-431-56457-7_18",

language = "English",

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volume = "183",

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booktitle = "Mathematical Fluid Dynamics, Present and Future",

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T1 - Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations

AU - Notsu, Hirofumi

AU - Tabata, Masahisa

PY - 2016

Y1 - 2016

N2 - Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitkäranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

AB - Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi– Pitkäranta's stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth's (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

KW - Error estimates

KW - Navier-Strokes equations

KW - Second-order scheme

KW - Stabilized Lagrange-Galerkin scheme

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U2 - 10.1007/978-4-431-56457-7_18

DO - 10.1007/978-4-431-56457-7_18

M3 - Conference contribution

AN - SCOPUS:85009786158

SN - 9784431564553

VL - 183

SP - 497

EP - 530

BT - Mathematical Fluid Dynamics, Present and Future

PB - Springer New York LLC

T2 - 8th CREST-SBM nternational Conference on Mathematical Fluid Dynamics, Present and Future, 2014

Y2 - 11 November 2014 through 14 November 2014

ER -

Error estimates of a stabilized lagrange–galerkin scheme of second-order in time for the navier–stokes equations

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