Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions

Abstract : We derive H(curl)-error estimates and improved L 2-error estimates for the Maxwell equations. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than 1 2 and can be arbitrarily close to 0 when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in H(curl)-and H(div)-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev regularity index arbitrarily close to 0. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.
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Computers and Mathematics with Applications, Elsevier, 2017, 75 (3), pp.967-983. 〈10.1016/j.camwa.2017.10.017〉
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Alexandre Ern, Jean-Luc Guermond. Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions. Computers and Mathematics with Applications, Elsevier, 2017, 75 (3), pp.967-983. 〈10.1016/j.camwa.2017.10.017〉. 〈hal-01531940v2〉

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