On commuting $p$-version projection-based interpolation on tetrahedra

Abstract

On the reference tetrahedron $\hat{K}$, we define three projection-based interpolation operators on $H^2(\hat{K})$, ${\mathbf{H}}^1(\hat{K},\mathrm{curl})$, and ${\mathbf{H}}^1(\hat{K},\mathrm{div})$. These operators are projections onto space of polynomials, they have the commuting diagram property and feature the optimal convergence rate as the polynomial degree increases in $H^{1-s}(\hat{K})$, ${\mathbf{H}}^{-s}(\hat{K},\mathrm{curl})$, ${\mathbf{H}}^{-s}(\hat{K},\mathrm{div})$ for $0\le s\le 1$.