Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

D. Harutyunyan, F. Izsak, Jacobus J.W. van der Vegt, Mikhail A. Bochev

  • 9 Citations

Abstract

We consider an implicit a posteriori error estimation technique for the adaptive solution of the Maxwell equations with Nédélec edge finite element methods on three-dimensional domains. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. The discrete bilinear form of the local problems is shown to satisfy an inf–sup condition which ensures the well posedness of the error equations. An adaptive solution algorithm is developed based on the obtained error estimates. The performance of the method is tested on various problems including non-convex domains with non-smooth boundaries. The numerical results show that the estimated error, computed by the implicit a posteriori error estimation technique, correlates well with the actual error. On the meshes generated adaptively with the help of the error estimator, the achieved accuracy is higher than on globally refined meshes with comparable number degrees of freedom. Moreover, the rate of the error convergence on the locally adapted meshes is faster than that on the globally refined meshes.
Original languageUndefined
Article number10.1016/j.cma.2007.12.006
Pages (from-to)1620-1638
Number of pages19
JournalComputer methods in applied mechanics and engineering
Volume197
Issue number17-18
DOIs
StatePublished - Mar 2008

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Error analysis
Maxwell equations
Finite element method

Keywords

  • EWI-12005
  • MSC-74S05
  • MSC-65N15
  • IR-62193
  • MSC-65N50
  • METIS-250884
  • MSC-65N30

Cite this

Harutyunyan, D.; Izsak, F.; van der Vegt, Jacobus J.W.; Bochev, Mikhail A. / Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates.

In: Computer methods in applied mechanics and engineering, Vol. 197, No. 17-18, 10.1016/j.cma.2007.12.006, 03.2008, p. 1620-1638.

Research output: Scientific - peer-reviewArticle

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abstract = "We consider an implicit a posteriori error estimation technique for the adaptive solution of the Maxwell equations with Nédélec edge finite element methods on three-dimensional domains. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. The discrete bilinear form of the local problems is shown to satisfy an inf–sup condition which ensures the well posedness of the error equations. An adaptive solution algorithm is developed based on the obtained error estimates. The performance of the method is tested on various problems including non-convex domains with non-smooth boundaries. The numerical results show that the estimated error, computed by the implicit a posteriori error estimation technique, correlates well with the actual error. On the meshes generated adaptively with the help of the error estimator, the achieved accuracy is higher than on globally refined meshes with comparable number degrees of freedom. Moreover, the rate of the error convergence on the locally adapted meshes is faster than that on the globally refined meshes.",
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Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates. / Harutyunyan, D.; Izsak, F.; van der Vegt, Jacobus J.W.; Bochev, Mikhail A.

In: Computer methods in applied mechanics and engineering, Vol. 197, No. 17-18, 10.1016/j.cma.2007.12.006, 03.2008, p. 1620-1638.

Research output: Scientific - peer-reviewArticle

TY - JOUR

T1 - Adaptive finite element techniques for the Maxwell equations using implicit a posteriori error estimates

AU - Harutyunyan,D.

AU - Izsak,F.

AU - van der Vegt,Jacobus J.W.

AU - Bochev,Mikhail A.

N1 - Please note different possible spellings of the last author's name: "Botchev" or "Bochev".

PY - 2008/3

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N2 - We consider an implicit a posteriori error estimation technique for the adaptive solution of the Maxwell equations with Nédélec edge finite element methods on three-dimensional domains. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. The discrete bilinear form of the local problems is shown to satisfy an inf–sup condition which ensures the well posedness of the error equations. An adaptive solution algorithm is developed based on the obtained error estimates. The performance of the method is tested on various problems including non-convex domains with non-smooth boundaries. The numerical results show that the estimated error, computed by the implicit a posteriori error estimation technique, correlates well with the actual error. On the meshes generated adaptively with the help of the error estimator, the achieved accuracy is higher than on globally refined meshes with comparable number degrees of freedom. Moreover, the rate of the error convergence on the locally adapted meshes is faster than that on the globally refined meshes.

AB - We consider an implicit a posteriori error estimation technique for the adaptive solution of the Maxwell equations with Nédélec edge finite element methods on three-dimensional domains. On each element of the tessellation an equation for the error is formulated and solved with a properly chosen local finite element basis. The discrete bilinear form of the local problems is shown to satisfy an inf–sup condition which ensures the well posedness of the error equations. An adaptive solution algorithm is developed based on the obtained error estimates. The performance of the method is tested on various problems including non-convex domains with non-smooth boundaries. The numerical results show that the estimated error, computed by the implicit a posteriori error estimation technique, correlates well with the actual error. On the meshes generated adaptively with the help of the error estimator, the achieved accuracy is higher than on globally refined meshes with comparable number degrees of freedom. Moreover, the rate of the error convergence on the locally adapted meshes is faster than that on the globally refined meshes.

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