High-frequency bounds for the Helmholtz equation under parabolic trapping and applications in numerical analysis

TL;DR

This paper establishes the first polynomial-loss resolvent estimate for the Helmholtz equation under parabolic trapping obstacles, enabling wavenumber-explicit convergence proofs for numerical methods in such trapping scenarios.

Contribution

It introduces a novel resolvent estimate for parabolic trapping obstacles, expanding understanding beyond elliptic and hyperbolic trapping cases.

Findings

Proves polynomial-loss resolvent estimate for parabolic trapping.

Derives wavenumber-explicit convergence results for numerical methods.

Extends vector-field/multiplier techniques to trapping configurations.

Abstract

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realised through some sequence of wavenumbers, (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in 2-d), or two cubes (in 3-d), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral-equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analysing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle.