On a polyharmonic Dirichlet problem and boundary effects in surface spline approximation

TL;DR

This paper introduces an improved surface spline approximation scheme for smooth domains that effectively mitigates boundary effects by using an integral identity derived from boundary layer potentials, achieving precise approximation orders.

Contribution

It presents a novel approximation scheme that overcomes boundary effects in surface spline approximation using a minimal boundary layer potential approach.

Findings

Achieves precise $L_p$ approximation orders on smoothness spaces.

Effectively mitigates boundary effects with increased center density near boundaries.

Provides an integral identity linked to the Dirichlet problem for the $m$-fold Laplacian.

Abstract

For compact domains with smooth boundaries, we present an approximation scheme for surface spline approximation that delivers precise $L_p$ approximation orders on well known smoothness spaces. This scheme overcomes the boundary effects when centers are placed with greater density near to the boundary. It owes its success to an integral identity using a minimal number of boundary layer potentials, which, in turn is derived from the boundary layer potential solution to the Dirichlet problem for the $m$-fold Laplacian. Furthermore, his integral identity is shown to be the "native space extension" of the target function.