Optimal error estimation for H(curl)-conforming p-interpolation in two dimensions

TL;DR

This paper establishes an optimal error estimate for H(curl)-conforming p-interpolation operators in two dimensions, applicable to vector fields with arbitrary regularity, and extends the results to H(div)-conforming settings relevant for Maxwell's equations.

Contribution

It provides the first optimal error bounds for H(curl)-conforming p-interpolation in 2D and extends the analysis to H(div)-conforming cases for high-order boundary element methods.

Findings

Proves optimal error estimates for H(curl)-conforming p-interpolation.

Extends the results to H(div)-conforming settings.

Applicable to vector fields with arbitrary regularity r>0.

Abstract

In this paper we prove an optimal error estimate for the H(curl)-conforming projection based p-interpolation operator introduced in [L. Demkowicz and I. Babuska, p interpolation error estimates for edge finite elements of variable order in two dimensions, SIAM J. Numer. Anal., 41 (2003), pp. 1195-1208]. This result is proved on the reference element (either triangle or square) K for regular vector fields in H^r(curl,K) with arbitrary r>0. The formulation of the result in the H(div)-conforming setting, which is relevant for the analysis of high-order boundary element approximations for Maxwell's equations, is provided as well.