Closed-Form Exact Inverses of the Weakly Singular and Hypersingular Operators On Disks

TL;DR

This paper derives explicit closed-form inverses for weakly singular and hypersingular boundary operators on disks, enabling improved preconditioning in boundary element methods through new Calderón identities.

Contribution

It introduces exact inverse operators for boundary integral operators on disks, with explicit formulas and proofs of their properties, advancing boundary element method theory.

Findings

Explicit closed-form inverses for boundary operators on disks.

Proofs of continuity and ellipticity of inverse operators.

New Calderón-type identities for operator preconditioning.

Abstract

We introduce new boundary integral operators which are the exact inverses of the weakly singular and hypersingular operators for the Laplacian on flat disks. Moreover, we provide explicit closed forms for them and prove the continuity and ellipticity of their corresponding bilinear forms in the natural Sobolev trace spaces. This permit us to derive new Calder\'on-type identities that can provide the foundation for optimal operator preconditioning in Galerkin boundary element methods.