We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Pad\'e approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e. away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.

翻译：我们考虑通过求解对应边值问题来逼近无陷障碍物Helmholtz外部Dirichlet问题的解（边界数据来自平面波入射），其中外部区域被截断，且在人造边界上施加基于Dirichlet-to-Neumann映射Padé近似（任意阶）的局部吸收边界条件（最简单情形为阻抗边界条件）。我们给出了该近似在全域及障碍物固定邻域内（即远离人造边界处）相对误差的上界与下界。对于任意高频、固定的人造边界，我们的界值表明相对误差恒大于零且与频率无关，同时不受人造边界几何形状的影响。