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C 0 elements for generalized indefinite Maxwell equations

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Abstract

In this paper we develop the C 0 finite element method for a generalized curlcurl-grad div indefinite Maxwell problem in a Lipschitz domain such as nonconvex polygon for which the solution of the problem may be nonsmooth and only have the H r regularity for some r < 1. The ingredients of our method are that two ‘mass-lumping’ L 2 projectors are applied to curl and div operators in the problem and that C 0 linear element or isoparametric bilinear element enriched with one element-bubble on every triangle element or with two-element-bubbles on every quadrilateral element, respectively, is employed for each component of the nonsmooth solution. Due to the fact that the element-bubbles can be statically eliminated at element levels, our method is essentially three-nodes or four-nodes C 0 Lagrange element method. With two Fortin-type interpolations established, a very technical duality argument is elaborated to estimate the error for the indefinite problem. For the nonsmooth solution having the H r regularity where r may vary in the interval [0, 1), we obtain the error bound \({{\mathcal O}(h^r)}\) in an energy norm. Some numerical experiments are performed to confirm the theoretical error bounds.

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Authors and Affiliations

  1. School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China

    Huoyuan Duan

  2. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK

    Ping Lin

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Singapore

    Roger C. E. Tan

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  1. Huoyuan Duan
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  2. Ping Lin
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  3. Roger C. E. Tan
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Correspondence to Huoyuan Duan.

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Duan, H., Lin, P. & Tan, R.C.E. C 0 elements for generalized indefinite Maxwell equations. Numer. Math. 122, 61–99 (2012). https://doi.org/10.1007/s00211-012-0456-x

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  • DOI: https://doi.org/10.1007/s00211-012-0456-x

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