Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation of layer potentials involving the free space Green's function. However, it seems to work in general unstructured geometry, and Galerkin finite element discretization leads to convergence under the usual mesh constraints imposed by G{\aa}rding-type inequalities. The nonlocal boundary conditions are readily approximated by fast multipole methods, and the resulting linear system can be preconditioned by the purely local operator involving transmission boundary conditions.

翻译：外海海尔姆霍尔茨问题的数字解析要求用某种方法来解决域间脱轨问题。作为近似不反映边界条件和援引迪里赫莱特至尼乌曼地图的替代办法,我们引入了新的非本地边界条件。这一条件十分准确,需要评估自由空间绿化功能的层潜力。然而,它似乎在一般情况下不结构的几何学上起作用,而加勒金的有限元素分解导致在G~a}拉动型不平等的通常网状限制下趋同。非本地边界条件很容易被快速多极方法所近,由此产生的线性系统可以由纯当地经营者在传输边界条件上设定先决条件。