High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

Abstract

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L 2 (Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp, and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is curved. Together, these results give bounds on the condition number of the operator on L 2 (Γ); this is the first time L 2 (Γ) condition-number bounds have been proved for this operator for obstacles other than balls.

Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

“…By the bound (1.14) from Assumption A0 and then (2.14), [45] by expressing (B k,reg ) −1 in terms of appropriate exterior and interior solution operators using [15, Lemma 6.1, Equation 83], and then using the bounds on the exterior Neumann problem from [22]/ [113] and the relevant interior problem (an interior impedance-like problem involving S ik in the boundary condition) from the combination of [43,Theorem 4.6] and [44,Lemma 3.2].…”

“…Similarly, Fig. 22 shows B k,reg L 2 ( )→L 2 ( ) being essentially constant for the range of k considered, although, at least in 2-d, B k,reg L 2 ( )→L 2 ( ) k 1/4 for large enough k; indeed, [26,Theorem 4.6] shows that D k L 2 ( )→L 2 ( ) k 1/4 for a certain class of 2-d domains (to see that the elliptic cavity falls in this class, take the points x 1 and x 2 in the statement of [26,Theorem 4.6] to lie on one of the flat ends of the cavity, with x 2 in the middle of this end, and x 1 at one of the corners) and [45,Theorems 4.6 and 4.8] Regarding the top-right plots: these show both (i) the feature F2, i.e. that while the norms of the inverses of the boundary-integral operators grow exponentially through k e m,0 , and thus the smallest singular values should decrease exponentially, this growth/decay stagnates, and (ii) that the smallest eigenvalue modulus is very close the smallest singular value, giving indirect evidence for Assumption A2, i.e.…”