The hp discontinuous Galerkin method for delay differential equations with nonlinear vanishing delay

Qiumei Huang, Hehu Xie, Hermann Brunner

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We present the $hp$-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the $hp$ discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.




LanguageEnglish
PagesA1604-A1620
Number of pages17
JournalSIAM Journal on Scientific Computing
Volume35
Issue number3
DOIs
Publication statusPublished - 2013

Fingerprint

Discontinuous Galerkin Method
Galerkin methods
Delay Differential Equations
Differential equations
Graded Meshes
Hp-version
Regularity Properties
Smooth Solution
Error Bounds
Rate of Convergence
Exact Solution
Polynomials
Numerical Solution
Mesh
Numerical Examples
Polynomial

Keywords

  • hp
  • discontinuous galerkin method
  • delay
  • differential equations
  • nonlinear vanishing delay

Cite this

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The hp discontinuous Galerkin method for delay differential equations with nonlinear vanishing delay. / Huang, Qiumei; Xie, Hehu; Brunner, Hermann.

In: SIAM Journal on Scientific Computing, Vol. 35, No. 3, 2013, p. A1604-A1620.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The hp discontinuous Galerkin method for delay differential equations with nonlinear vanishing delay

AU - Huang, Qiumei

AU - Xie, Hehu

AU - Brunner, Hermann

PY - 2013

Y1 - 2013

N2 - We present the $hp$-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the $hp$ discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.

AB - We present the $hp$-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the $hp$ discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.

KW - hp

KW - discontinuous galerkin method

KW - delay

KW - differential equations

KW - nonlinear vanishing delay

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M3 - Article

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SP - A1604-A1620

JO - SIAM Journal on Scientific Computing

T2 - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

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