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Computation of resonance frequencies for Maxwell equations in non-smooth domains

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Topics in Computational Wave Propagation

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 31))

Summary

We address the computation by finite elements of the non-zero eigenvalues of the (curl, curl) bilinear form with perfect conductor boundary conditions in a polyhedral cavity. One encounters two main difficulties: (i) The infinite dimensional kernel of this bilinear form (the gradient fields), (ii) The unbounded singularities of the eigen-fields near corners and edges of the cavity. We first list possible variational spaces with their functional properties and provide a short description of the edge and corner singularities. Then we address different formulations using a Galerkin approximation by edge elements or nodal elements.

After a presentation of edge elements, we concentrate on the functional issues connected with the use of nodal elements. In the framework of conforming methods, nodal elements are mandatory if one regularises the bilinear form (curl, curl) in order to get rid of the gradient fields. A plain regularisation with the (div, div) bilinear form converges to a wrong solution if the domain has reentrant edges or corners. But remedies do exist We will present the method of addition of singular functions, and the method of regularisation with weight, where the (div, div) bilinear form is modified by the introduction of a weight which can be taken as the distance to reentrant edges or corners.

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Authors and Affiliations

  1. IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes, France

    Martin Costabel & Monique Dauge

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  1. Martin Costabel
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  2. Monique Dauge
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Editors and Affiliations

  1. Department of Mathematics, University of Strathclyde, Richmond Street 26, Glasgow, G1 1XH, UK

    Mark Ainsworth  & Penny Davies  & 

  2. Department of Mathematics, Heriot-Watt University, Riccarton Campus, EH14 4AS, Edinburgh, UK

    Dugald Duncan  & Bryan Rynne  & 

  3. Department of Mathematics and Computer Sciences, Colorado School of Mines, 80401-1887, Golden, USA

    Paul Martin

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Costabel, M., Dauge, M. (2003). Computation of resonance frequencies for Maxwell equations in non-smooth domains. In: Ainsworth, M., Davies, P., Duncan, D., Rynne, B., Martin, P. (eds) Topics in Computational Wave Propagation. Lecture Notes in Computational Science and Engineering, vol 31. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55483-4_4

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  • DOI: https://doi.org/10.1007/978-3-642-55483-4_4

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  • Print ISBN: 978-3-540-00744-9

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