A new H(div)-conforming p-interpolation operator in two dimensions

TL;DR

This paper introduces a new H(div)-conforming p-interpolation operator for 2D elements that is stable, respects the commuting diagram property, and is suitable for electromagnetic boundary integral problems.

Contribution

It constructs a novel projection-based interpolation operator that requires minimal regularity assumptions and is applicable to both triangular and square reference elements.

Findings

Operator is stable with respect to polynomial degrees.

Satisfies the commuting diagram property.

Provides an error estimate in the $ ilde H^{-1/2}(div,K)$ norm.

Abstract

In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only $H^r(K) \cap \tilde H^{-1/2}(div,K)$-regularity (r > 0) on the reference element K (either triangle or square). We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space $\tilde H^{-1/2}(div,K)$, which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.