Interpolation Operator on negative Sobolev Spaces

TL;DR

This paper introduces a Scott--Zhang type projection operator compatible with negative Sobolev spaces, enabling optimal convergence rates and applications in parabolic problems and finite element methods.

Contribution

It presents a novel projection operator that is stable in negative Sobolev norms and extends interpolation theory to these spaces.

Findings

Operator is stable in negative Sobolev norms

Achieves optimal convergence rates

Enables improved finite element error estimates

Abstract

We introduce a Scott--Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method.