A superconvergent finite element method for the korteweg-de vries equation

Douglas N. Arnold, Ragnar Winther

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.

Original languageEnglish (US)
Pages (from-to)23-36
Number of pages14
JournalMathematics of Computation
Volume38
Issue number157
DOIs
StatePublished - Jan 1 1982

Fingerprint

Korteweg-de Vries equation
Discrete Equations
Korteweg-de Vries Equation
Finite Element Method
Finite element method
Forcing Term
Superconvergence
Unconditionally Stable
Parabolic Problems
Perturbation
Term
Vertex of a graph
Estimate
Form

Cite this

A superconvergent finite element method for the korteweg-de vries equation. / Arnold, Douglas N.; Winther, Ragnar.

In: Mathematics of Computation, Vol. 38, No. 157, 01.01.1982, p. 23-36.

Research output: Contribution to journalArticle

Arnold, Douglas N. ; Winther, Ragnar. / A superconvergent finite element method for the korteweg-de vries equation. In: Mathematics of Computation. 1982 ; Vol. 38, No. 157. pp. 23-36.
@article{04837ef44fa842bbbdeb07886fec8a76,
title = "A superconvergent finite element method for the korteweg-de vries equation",
abstract = "An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.",
author = "Arnold, {Douglas N.} and Ragnar Winther",
year = "1982",
month = "1",
day = "1",
doi = "10.1090/S0025-5718-1982-0637284-8",
language = "English (US)",
volume = "38",
pages = "23--36",
journal = "Mathematics of Computation",
issn = "0025-5718",
publisher = "American Mathematical Society",
number = "157",

}

TY - JOUR

T1 - A superconvergent finite element method for the korteweg-de vries equation

AU - Arnold, Douglas N.

AU - Winther, Ragnar

PY - 1982/1/1

Y1 - 1982/1/1

N2 - An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.

AB - An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.

UR - http://www.scopus.com/inward/record.url?scp=84968517899&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968517899&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1982-0637284-8

DO - 10.1090/S0025-5718-1982-0637284-8

M3 - Article

VL - 38

SP - 23

EP - 36

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 157

ER -