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We estimate best-approximation errors using vector-valued finite elements forfields with low regularity in the scale of fractional-order Sobolev spaces. Byassuming additionally that the target field has a curl or divergence property,we establish upper bounds on these errors that can be localized to the meshcells. These bounds are derived using the quasi-interpolation errors with orwithout boundary prescription derived in [A. Ern and J.-L. Guermond, ESAIMMath. Model. Numer. Anal., 51 (2017), pp.~1367--1385]. By using theface-to-cell lifting operators analyzed in [A. Ern and J.-L. Guermond, Found.Comput. Math., (2021)], and exploiting the additional assumption made on thecurl or the divergence of the target field, a localized upper bound on thequasi-interpolation error is derived. As an illustration, we show how to applythese results to the error analysis of the curl-curl problem associated withMaxwell's equations.

Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence