We propose an Edge Multiscale Finite Element Method (EMsFEM) based on an Interior Penalty Discontinuous Galerkin (IPDG) formulation for the heterogeneous Helmholtz problems with large wavenumber. A novel local multiscale space is constructed by solving local problems with a mixed boundary condition composed of a nonhomogeneous Dirichlet boundary condition and an absorbing boundary condition, which can capture the local behavior of the wave propagation and local media information. The key ingredient of our method consists of choosing appropriate Dirichlet data inspired by recent development on edge multiscale basis functions [9], [24], [39], [49]. An IPDG formulation is applied to facilitate generating a sparse linear system and to reduce computational complexity. The convergence rate is derived for wavelet-based and polynomial-based edge multiscale basis functions. Extensive numerical tests in two and three dimensional heterogeneous media are presented to show the supreme performance of our method.

我们提出了一种基于内部惩罚不连续伽勒金（IPDG）公式的边缘多尺度有限元方法（EMsFEM），用于大波数异质亥姆霍兹问题。通过求解由非均匀狄利克雷边界条件和吸收边界条件组成的混合边界条件解决局部问题，构造了一种新颖的局部多尺度空间，可以捕获波传播的局部行为和局部介质信息。我们方法的关键组成部分包括选择合适的Dirichlet数据，该数据受边缘多尺度基函数[9]，[24]，[39]，[49]的最新发展启发。采用IPDG公式有助于生成稀疏线性系统并降低计算复杂度。对于基于小波和多项式的边缘多尺度基函数，得出收敛速度。提出了在二维和三维异质介质中的大量数值测试，以显示我们方法的最高性能。