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 Padé 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.
Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations
研究了无界空间区域上一维和二维非局部波动方程高阶数值逼近的稳定性和收敛性。首先用基于正交的有限差分格式离散空间非局部算子，然后用显式差分格式逼近时间导数，得到一个完全离散的无穷系统.在此基础上，构造了Dirichlet-to-Neumann（DtN）型吸收边界条件（ABCs），将无限大离散系统化为有限大离散系统。为此，我们首先采用[Du，Zhang and Zheng，\emph{Commun. Comput. Phys.}，24（4）：1049--1072，2018 and Du，Han，Zhang and Zheng，\emph{SIAM J. Sci. Comp.}，40（3）：A1430--A1445，2018]中的思想，分别导出一维和二维情况下的Dirichlet到Dirichlet（DtD）型映射。然后利用离散非局部Green第一恒等式从DtD型映射得到离散DtN型映射。由此产生的DtN型映射使得有可能进行减少的问题的稳定性和收敛性分析。数值实验证明了该方法的准确性和有效性。
