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We derive $H_{\text{curl}}$-error estimates and improved $L^2$-errorestimates for the Maxwell equations approximated using edge finite elements.These estimates only invoke the expected regularity pickup of the exactsolution in the scale of the Sobolev spaces, which is typically lower than$\frac12$ and can be arbitrarily close to $0$ when the material properties areheterogeneous. The key tools for the analysis are commuting quasi-interpolationoperators in $H_{\text{curl}}$- and $H_{\text{div}}$-conforming finite elementspaces and, most crucially, newly-devised quasi-interpolation operatorsdelivering optimal estimates on the decay rate of the best-approximation errorfor functions with Sobolev smoothness index arbitrarily close to $0$. Theproposed analysis entirely bypasses the technique known in the literature asthe discrete compactness argument.

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