Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

Samir Karaa, Amiya K. Pani

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)-norm are derived again under minimal smoothness on initial data.

Original languageEnglish
Pages (from-to)113-134
Number of pages22
JournalJournal of Computational and Applied Mathematics
Volume275
DOIs
Publication statusPublished - 2014

Fingerprint

Integrodifferential equations
Optimal Error Estimates
Mixed Methods
Hyperbolic Equations
Integro-differential Equation
Smoothness
Norm
Finite element method
Mixed Finite Element Method
Implicit Scheme
Error Estimates
Formulation
Energy
Estimate
Class
Standards

Keywords

  • Completely discrete implicit method
  • Hyperbolic integro-differential equation
  • Minimal smoothness on initial data
  • Mixed finite element method
  • Optimal error estimates
  • Semidiscrete Galerkin approximation

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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abstract = "In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)-norm are derived again under minimal smoothness on initial data.",
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T1 - Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

AU - Karaa, Samir

AU - Pani, Amiya K.

PY - 2014

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N2 - In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)-norm are derived again under minimal smoothness on initial data.

AB - In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L∞(L2)-error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L∞(L∞)-norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L∞(L2)-norm are derived again under minimal smoothness on initial data.

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KW - Hyperbolic integro-differential equation

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KW - Semidiscrete Galerkin approximation

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