求解Helmholtz方程的快速算法

A Fast Propagation Method for the Helmholtz

Equation?

LENG Wei

【摘 要】Abstract:A fast method is proposed for solving the high frequency Helmholtz equation.The building block of the new fast method is an overlapping domain decomposition method for layered medium.In the new fast method,the computation domain is fi rstly decomposed hierarchically into many subdomains on di ff erent levels.Then the mapping from incident waves to out-going waves on all the subdomains are set up.Finally,the wave propagates on the subdomain boundaries on di ff erent levels to reach the solution to the Helmholtz equation.The new fast method is of low complexity,and suitable for parallel computing.Numerical experiments show that with the new fast method,2D Helmholtz equations with half billion unknowns could be solved efficiently on massively parallel machines. 【期刊名称】工程数学学报 【年(卷),期】2015(000)005 【总页数】17

【关键词】Keywords:Helmholtz equation; fi nite di ff erence;fast method;domain decomposition method;PML

【文献来源】https://www.zhangqiaokeyan.com/academic-journal-cn_chinese-journal-engineering-mathematics_thesis/0201211810105.html

1 Introduction

We consider in this paper to solve the Helmholtz equation in the full spaceR2,with Sommerfeld radiation condition where k is the wave number.

Many domain decomposition method have recently been developed to solve the Helmholtz equation,most of them are non-overlapped,and the major differences between them are the interface conditions.Engquist and Ying[1,2]proposed a sweepingpreconditioner by approximating the inverse of Schur complements in the LDLtfactorization,Stolk[3]proposed a domain decomposition method with a transmission condition based on perfect matched layers,Vion and Geuzaine[4]proposed a double sweep preconditioner that use a transmission condition that involves Dirichlet-to-Neumann(DtN)operator,Zepeda and Demanet[5]introduced the method of polarized trace that use a transmission condition in boundary integral form,Liu and Ying[6]developed an additive sweeping preconditioner that use a transmission condition built with the boundary values of the intermediate wave directly.Chen and Xiang[7,8]proposed the source transfer domain decomposition method that transfer the source in subdomains,and recently Du and Wu[9]improved the method so that the transfer applies in both directions.

The domain decomposition methods in the literature usually

approximately solve the Helmholtz equation with varying

medium,either with approximated interface condition or with approximated Green function,thus they are commonly used as preconditioners for Krylov subspace method such as GMRES.

An overlapping source transfer domain decomposition method is proposed for Helmholtz equation with layered medium,the method follows the natural wave traveling process in layered medium,which involves the ref l ections and refractions at the interface of the layers.The convergence of the new domain decomposition method is ensured by the overlapping region,and the good accuracy of the new domain decomposition method makes it the building block of the new fast method.

The domain decomposition method suf f ers from slow convergence rate when the number of subdomains is large,thus multilevel grid is needed so that the information is brought to far away subdomains without passing the subdomains on the way.The upper level grid for Poisson type problem could be coarser since the amount of information decreases

fast

as

the

distance

grows.However,for

Helmholtz

equation,the grid size should be maintained small to represent wave shapes on the upper level grid.Fortunately,the trace on the subdomain boundaries could be used to represent the solution on the subdomain,thus the computation cost on upper level grid is not