Numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations

Ahmed H Elsheikh, S Smith, S E Chidiac

Research output: Contribution to journalArticle

Abstract

A numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations is presented. The estimator used is based on the solution of local problems subjected to Neumann boundary conditions. The estimated errors are calculated in a weighted energy norm, a stability norm and an approximate fractional order norm in order to study the effect of the error norm on both the effectivity index of the estimated errors and the mesh adaptivity process. The reported numerical results are in general better than what is available in the literature. The results reveal that the reliability of the estimated errors depends on the relation between the mesh size and the size of local features in the solution. The stability norm is found to have some advantages over the weighted energy norm in terms of producing effectivity indices closer to the optimal unit value, especially for problems with internal sharp layers. Meshes adapted by the element residual method measured in the stability norm conform to the sharp layers and are shown to be less dependent on the wind direction.

Original languageEnglish
Pages (from-to)711-726
Number of pages16
JournalCommunications in Numerical Methods in Engineering
Volume24
Issue number9
Early online date19 Jan 2007
DOIs
Publication statusPublished - Sep 2008

Fingerprint

A Posteriori Error Estimation
Advection-diffusion Equation
Advection
Numerical Investigation
Error analysis
Norm
Mesh Adaptivity
Mesh
Boundary conditions
Local Features
Fractional Order
Energy
Neumann Boundary Conditions
Internal
Estimator
Numerical Results
Unit
Dependent

Keywords

  • A posteriori error estimates
  • Adaptive finite element
  • Advection-diffusion equations
  • Error norm

ASJC Scopus subject areas

  • Applied Mathematics
  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Software
  • Engineering(all)

Cite this

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abstract = "A numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations is presented. The estimator used is based on the solution of local problems subjected to Neumann boundary conditions. The estimated errors are calculated in a weighted energy norm, a stability norm and an approximate fractional order norm in order to study the effect of the error norm on both the effectivity index of the estimated errors and the mesh adaptivity process. The reported numerical results are in general better than what is available in the literature. The results reveal that the reliability of the estimated errors depends on the relation between the mesh size and the size of local features in the solution. The stability norm is found to have some advantages over the weighted energy norm in terms of producing effectivity indices closer to the optimal unit value, especially for problems with internal sharp layers. Meshes adapted by the element residual method measured in the stability norm conform to the sharp layers and are shown to be less dependent on the wind direction.",
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Numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations. / Elsheikh, Ahmed H; Smith, S; Chidiac, S E.

In: Communications in Numerical Methods in Engineering, Vol. 24, No. 9, 09.2008, p. 711-726.

Research output: Contribution to journalArticle

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AU - Smith, S

AU - Chidiac, S E

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AB - A numerical investigation of the reliability of a posteriori error estimation for advection-diffusion equations is presented. The estimator used is based on the solution of local problems subjected to Neumann boundary conditions. The estimated errors are calculated in a weighted energy norm, a stability norm and an approximate fractional order norm in order to study the effect of the error norm on both the effectivity index of the estimated errors and the mesh adaptivity process. The reported numerical results are in general better than what is available in the literature. The results reveal that the reliability of the estimated errors depends on the relation between the mesh size and the size of local features in the solution. The stability norm is found to have some advantages over the weighted energy norm in terms of producing effectivity indices closer to the optimal unit value, especially for problems with internal sharp layers. Meshes adapted by the element residual method measured in the stability norm conform to the sharp layers and are shown to be less dependent on the wind direction.

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