A heterogeneous FMM for layered media Helmholtz equation I: Two layers in R-2

作者：Cho Min Hyung; Huang Jingfang; Chen Dangxing; Cai Wei^*

来源：Journal of Computational Physics, 2018, 369: 237-251.

DOI：10.1016/j.jcp.2018.05.007

• Helmholtz equation
• Impedance boundary condition
• Fast multipole method
• Hierarchical model
• Low-rank representation
• Multi-layered media

In this paper, we introduce a new heterogeneous fast multipole method (H-FMM) for 2-D Helmholtz equation in layered media. To illustrate the main algorithm ideas, we focus on the case of two layers in this work. The key compression step in the H-FMM is based on a fact that the multipole expansion for the sources of the free-space Green's function can be used also to compress the far field of the sources of the layered-media or domain Green's function, and a similar result exists for the translation operators for the multipole and local expansions. The mathematical error analysis is shown rigorously by an image representation of the Sommerfeld spectral form of the domain Green's function. As a result, in the H-FMM algorithm, the multipole-to-multipole, multipole-to-local, and local-to-local translation operators are the same as those in the free-space case, allowing easy adaptation of existing free-space FMM. All the spatially variant information of the domain Green's function is collected into the multipole-to-local translations and therefore the FMM becomes heterogeneous. The compressed representation further reduces the cost of evaluating the domain Green's function when computing the local direct interactions. Preliminary numerical experiments are presented to demonstrate the efficiency and accuracy of the algorithm with much improved performance over some existing methods for inhomogeneous media.

• 出版日期2018-9-15