Local high-order regularization and applications to hp-methods

TL;DR

This paper introduces a local high-order regularization operator that enhances hp-methods by providing approximation and inverse estimates based on local regularity and mesh parameters, improving error analysis and interpolation.

Contribution

It develops a new regularization operator on $L_1$ with local approximation properties and combines it with hp-interpolation to create a minimal smoothness quasi-interpolation operator for hp-methods.

Findings

The regularization operator has approximation properties tied to local regularity and length scale.

The hp-Clément type quasi-interpolation operator requires minimal smoothness and maintains approximation accuracy.

Application to hp-boundary element methods yields explicit residual error estimates.

Abstract

We develop a regularization operator based on smoothing on a locally defined length scale. This operator is defined on $L_1$ and has approximation properties that are given by the local regularity of the function it is applied to and the local length scale. Additionally, the regularized function satisfies inverse estimates commensurate with the approximation orders. By combining this operator with a classical hp-interpolation operator, we obtain an hp-Cl\'ement type quasi-interpolation operator, i.e., an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree. As a second application, we consider residual error estimates in hp-boundary element methods that are explicit in the local mesh size and the local approximation order.