A priori error estimates for finite volume element approximations to second order linear hyperbolic integro-differential equations

Samir Karaa, Amiya K. Pani

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro-differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in L(L2) and L(H1) norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate is derived in L(L) norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence.

Original languageEnglish
Pages (from-to)401-429
Number of pages29
JournalInternational Journal of Numerical Analysis and Modeling
Volume12
Issue number3
Publication statusPublished - 2015

Fingerprint

Finite Volume Element
A Priori Error Estimates
Integrodifferential equations
Optimal Error Estimates
Hyperbolic Equations
Integro-differential Equation
Regularity
Finite Volume Element Method
Norm
Numerical Quadrature
Explicit Methods
Order of Convergence
Approximation
Numerical Experiment
Estimate
Experiments
Class

Keywords

  • Completely discrete scheme
  • Finite volume element
  • Hyperbolic integro-differential equation
  • Numerical quadrature
  • Optimal error estimates
  • Ritz-Volterra projection
  • Semidiscrete method

ASJC Scopus subject areas

  • Numerical Analysis

Cite this

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abstract = "In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro-differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in L∞(L2) and L∞(H1) norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate is derived in L∞(L∞) norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence.",
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AB - In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro-differential equations in a two-dimensional convex polygonal domain. The effect of numerical quadrature is also examined. In the semidiscrete case, optimal error estimates in L∞(L2) and L∞(H1) norms are shown to hold with minimal regularity assumptions on the initial data, whereas quasi-optimal estimate is derived in L∞(L∞) norm under higher regularity on the data. Based on a second order explicit method in time, a completely discrete scheme is examined and optimal error estimates are established with a mild condition on the space and time discretizing parameters. Finally, some numerical experiments are conducted which confirm the theoretical order of convergence.

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