A non-conforming domain decomposition approximation for the Helmholtz screen problem with hypersingular operator

TL;DR

This paper introduces a non-conforming domain decomposition method for the 3D Helmholtz hypersingular operator, demonstrating convergence and validating results through numerical experiments, especially for small wave numbers.

Contribution

It develops a novel non-conforming domain decomposition approach with Nitsche coupling for Helmholtz hypersingular operators, providing convergence analysis and numerical validation.

Findings

Method converges almost quasi-optimally for small wave numbers.

Numerical experiments confirm the theoretical error estimates.

Approach effectively handles unbounded exterior domain problems.

Abstract

We present and analyze a non-conforming domain decomposition approximation for a hypersingular operator governed by the Helmholtz equation in three dimensions. This operator appears when considering the corresponding Neumann problem in unbounded domains exterior to open surfaces. We consider small wave numbers and low-order approximations with Nitsche coupling across interfaces. Under appropriate assumptions on mapping properties of the weakly singular and hypersingular operators with Helmholtz kernel, we prove that this method converges almost quasi-optimally. Numerical experiments confirm our error estimate.