A numerical study on the stability of a class of Helmholtz problems

Sharp High-Frequency Estimates for the Helmholtz Equation and Applications to Boundary Integral Equations

“…The bound (19), used in conjunction with the recent results of Galkowski-Smith and Galkowski [28], [34], on essentially the norm of A ′ k,η , almost completes the study of the conditioning of A ′ k,η in the high-frequency limit, i.e., the study of…”

“…(2) In this paper we focus on the direct integral equation for the exterior Dirichlet problem, i.e., the equation where the unknown has an immediate physical meaning (in this case, it is the Neuman trace ∂ + n u) but an analogous bound to (19)…”

“…1, and so covered the case when |η| ∼ k and Ω − is star-shaped with respect to a ball, using the bound (21). Thanks to the bound (19), however, this analysis is now valid when Ω + is nontrapping and η satisfies Assumption 1.6. (Note that the error analysis of the hp-boundary element method conducted in [46], [52] only requires (A ′ k,η ) −1…”

Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

“…But compared with the second-and fourth-order schemes and the parameter one, the sixthorder method investigated here can achieve the best computational accuracy in all tested cases. Finally, we consider a practical model which is reduced from the large cavity electromagnetic scattering and has been investigated in [6,[34][35][36]. In this problem, Ω := (0, 1)×(0, 1 4 ), 1 4 ], Γ t := [1, 0] × { 1 4 }, Γ l := {0} × [ 1 4 , 0], f = 0, u = 0 on Γ b ∪ Γ r ∪ Γ l , ∂u ∂y + iku = g t on Γ t , which is the lowest-order approximation of the radiation boundary condition (see [6,18]).…”

“…Further research can be found in [14]. But in many cases, such as the Helmholtz equation after reduction from the large cavity electromagnetic scattering, the inhomogeneous Robin boundary condition (2) is necessary (see [6,[34][35][36]). Obviously, the high-order scheme in this case cannot be got following the process in [12] due to the existence of the function g. On the other hand, the Robin boundary condition is only imposed on a single boundary in [12,14].…”