Asymptotically exact Discontinuous Galerkin error estimates for linear symmetric hyperbolic systems

S. Adjerid, Thomas Weinhart

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

Abstract

We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p and p+1. We apply these asymptotic results to observe that projections of the error are pointwise O(hp+2)-superconvergent in some cases. Then we solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.
Original languageEnglish
Pages (from-to)101-131
Number of pages32
JournalApplied numerical mathematics
Volume76
DOIs
Publication statusPublished - 2014

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Symmetric Hyperbolic Systems
Discontinuous Galerkin
Error Estimates
Error analysis
A Posteriori Error Analysis
Electromagnetism
Legendre polynomial
Discretization Error
Systems of Partial Differential Equations
Smooth Solution
Hyperbolic Systems
Error Analysis
Linear Combination
Computational Results
Acoustics
Linear Systems
Projection
Finite Element
Partial differential equations
First-order

Keywords

  • IR-89123
  • METIS-301979

Cite this

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abstract = "We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p and p+1. We apply these asymptotic results to observe that projections of the error are pointwise O(hp+2)-superconvergent in some cases. Then we solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.",
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Asymptotically exact Discontinuous Galerkin error estimates for linear symmetric hyperbolic systems. / Adjerid, S.; Weinhart, Thomas.

In: Applied numerical mathematics, Vol. 76, 2014, p. 101-131.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Asymptotically exact Discontinuous Galerkin error estimates for linear symmetric hyperbolic systems

AU - Adjerid, S.

AU - Weinhart, Thomas

PY - 2014

Y1 - 2014

N2 - We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p and p+1. We apply these asymptotic results to observe that projections of the error are pointwise O(hp+2)-superconvergent in some cases. Then we solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.

AB - We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p and p+1. We apply these asymptotic results to observe that projections of the error are pointwise O(hp+2)-superconvergent in some cases. Then we solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.

KW - IR-89123

KW - METIS-301979

U2 - 10.1016/j.apnum.2013.06.007

DO - 10.1016/j.apnum.2013.06.007

M3 - Article

VL - 76

SP - 101

EP - 131

JO - Applied numerical mathematics

JF - Applied numerical mathematics

SN - 0168-9274

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