Local inverse estimates for non-local boundary integral operators

TL;DR

This paper establishes local inverse estimates for boundary integral operators related to the Laplace equation on Lipschitz domains, providing explicit bounds for polynomial spaces in 2D and 3D, with applications to boundary element error analysis.

Contribution

It introduces explicit local inverse estimates for non-local boundary integral operators on Lipschitz domains, enhancing a posteriori error estimation in boundary element methods.

Findings

Explicit inverse estimates for boundary integral operators in 2D and 3D.

Application to efficiency estimates in boundary element a posteriori error analysis.

Results applicable to piecewise polynomial ansatz spaces.

Abstract

We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded d-dimensional Lipschitz domain Omega for d >= 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d = 2 or 3, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a posteriori error estimation in boundary element methods is given.