Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations
Provides rigorous wavenumber-explicit bounds for boundary integral operators, crucial for analyzing and preconditioning numerical methods for high-frequency Helmholtz problems.
The paper derives sharp high-frequency estimates for solutions to the Helmholtz equation in interior and exterior domains, which yield bounds on the inverses of combined-field boundary integral operators. These estimates are explicit in the wavenumber and domain geometry.
We consider three problems for the Helmholtz equation in interior and exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems.