Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence
Abstract
We estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of fractional-order Sobolev spaces. By assuming additionally that the target field has a curl or divergence property, we establish upper bounds on these errors that can be localized to the mesh cells. These bounds are derived using the quasi-interpolation errors with or without boundary prescription derived in [A. Ern and J.-L. Guermond, ESAIM Math. Model. Numer. Anal., 51 (2017), pp.~1367--1385]. By using the face-to-cell lifting operators analyzed in [A. Ern and J.-L. Guermond, Found. Comput. Math., (2021)], and exploiting the additional assumption made on the curl or the divergence of the target field, a localized upper bound on the quasi-interpolation error is derived. As an illustration, we show how to apply these results to the error analysis of the curl-curl problem associated with Maxwell's equations.
Cite
@article{arxiv.2201.01708,
title = {Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence},
author = {Zhaonan Dong and Alexandre Ern and Jean-Luc Guermond},
journal= {arXiv preprint arXiv:2201.01708},
year = {2022}
}
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