论文标题
具有非本地边界条件的Helmholtz方程的有限元素
Finite elements for Helmholtz equations with a nonlocal boundary condition
论文作者
论文摘要
外部Helmholtz问题的数值分辨率需要某种域截断的方法。作为近似非反射边界条件的替代方案和Dirichlet到Neumann地图的调用，我们引入了一个新的非局部边界条件。这种情况是精确的，需要评估涉及自由空间绿色功能的层势。但是，它似乎在一般的非结构化几何形状中起作用，而Galerkin有限元离散化会导致在Gårding-type型不平等所施加的通常的网格约束下收敛。非局部边界条件很容易通过快速的多极方法近似，并且可以通过涉及传输边界条件的纯本地操作员对所得的线性系统进行预处理。
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 Gårding-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.
