Local high-order regularization and applications to hp-methods

We develop a regularization operator based on smoothing on a spatially varying length scale. This operator is defined for functions $u\in L^1$ and has approximation properties that are given by the local Sobolev regularity of $u$ and the local smoothing 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 $hp$ ‐Clément 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.