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

TL;DR

This paper introduces projection-based interpolation operators on a tetrahedron that preserve polynomial spaces, commute with differential operators, and achieve optimal convergence rates across various Sobolev spaces.

Contribution

It defines new projection operators on tetrahedral reference elements with commuting diagram properties and optimal convergence, extending the theory of $p$-version finite element methods.

Findings

Operators have optimal convergence rates as polynomial degree increases.

Operators preserve polynomial spaces and commute with differential operators.

Applicable to $H^2$, curl, and div spaces on tetrahedra.

Abstract

On the reference tetrahedron $\widehat K$, we define three projection-based interpolation operators on $H^2(\widehat K)$, ${\mathbf H}^1(\widehat K,\operatorname{\mathbf{curl}})$, and ${\mathbf H}^1(\widehat K,\operatorname{div})$. 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 $H^{1-s}(\widehat K)$, ${\mathbf H}^{-s}(\widehat K,\operatorname{\mathbf{curl}})$, ${\mathbf H}^{-s}(\widehat K,\operatorname{div})$ for $0 \leq s \leq 1$.