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

TL;DR

This paper derives localized error bounds for vector finite element approximations of fields with low regularity, especially those with curl or divergence properties, and applies these results to Maxwell's equations.

Contribution

It introduces new localized error estimates for vector-valued finite elements considering low regularity and curl/divergence constraints, extending previous quasi-interpolation techniques.

Findings

Localized error bounds depend on mesh cells and field properties.

Application to Maxwell's equations demonstrates practical relevance.

Enhanced understanding of approximation errors for low-regularity fields.

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.