A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations

Samir Karaa, Amiya K. Pani, Sangita Yadav

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, a priori hp-error estimates in L∞(L2)-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L∞(L2)-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

Original languageEnglish
Pages (from-to)1-23
Number of pages23
JournalApplied Numerical Mathematics
Volume96
DOIs
Publication statusPublished - May 6 2015

Fingerprint

Integrodifferential equations
Discontinuous Galerkin
Galerkin Approximation
Hyperbolic Equations
Integro-differential Equation
Galerkin methods
Estimate
Norm
A Priori Error Estimates
Discontinuous Galerkin Method
Polynomials
Order of Convergence
Volterra
Fluxes
Parabolic Problems
Elliptic Problems
Error Estimates
Exact Solution
Regularity
Numerical Experiment

Keywords

  • hp-Error estimates
  • Linear second order hyperbolic integro-differential equation
  • Local discontinuous Galerkin method
  • Mixed type Ritz-Volterra projection
  • Nonstandard formulation
  • Numerical experiments
  • Order of convergence
  • Role of stabilizing parameters
  • Semidiscrete and completely discrete schemes

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations. / Karaa, Samir; Pani, Amiya K.; Yadav, Sangita.

In: Applied Numerical Mathematics, Vol. 96, 06.05.2015, p. 1-23.

Research output: Contribution to journalArticle

@article{64a94c6ac9de47b8b1b9f89b600436c6,
title = "A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations",
abstract = "An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, a priori hp-error estimates in L∞(L2)-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L∞(L2)-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.",
keywords = "hp-Error estimates, Linear second order hyperbolic integro-differential equation, Local discontinuous Galerkin method, Mixed type Ritz-Volterra projection, Nonstandard formulation, Numerical experiments, Order of convergence, Role of stabilizing parameters, Semidiscrete and completely discrete schemes",
author = "Samir Karaa and Pani, {Amiya K.} and Sangita Yadav",
year = "2015",
month = "5",
day = "6",
doi = "10.1016/j.apnum.2015.04.006",
language = "English",
volume = "96",
pages = "1--23",
journal = "Applied Numerical Mathematics",
issn = "0168-9274",
publisher = "Elsevier",

}

TY - JOUR

T1 - A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations

AU - Karaa, Samir

AU - Pani, Amiya K.

AU - Yadav, Sangita

PY - 2015/5/6

Y1 - 2015/5/6

N2 - An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, a priori hp-error estimates in L∞(L2)-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L∞(L2)-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

AB - An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz-Volterra projection, a priori hp-error estimates in L∞(L2)-norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L∞(L2)-norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.

KW - hp-Error estimates

KW - Linear second order hyperbolic integro-differential equation

KW - Local discontinuous Galerkin method

KW - Mixed type Ritz-Volterra projection

KW - Nonstandard formulation

KW - Numerical experiments

KW - Order of convergence

KW - Role of stabilizing parameters

KW - Semidiscrete and completely discrete schemes

UR - http://www.scopus.com/inward/record.url?scp=84929151805&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84929151805&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2015.04.006

DO - 10.1016/j.apnum.2015.04.006

M3 - Article

VL - 96

SP - 1

EP - 23

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -