Some Overlapping Domain Decomposition Algorithms For 2-D Helmholtz Equation

With the development of technology and thorough research of various problems,people discover that the transmission problems in waveguide of the electromagnetic field, the elasticity waveguide problems and the diffraction problems in water wave of the ocean engineering are all based on Helmholtz equation. Therefore, how to solve the Helmholtz equation rapidly and accurately is becoming more and more important. It is difficult to solve the problem numerically, in particular, when the wave number k is very large. The overlapping domain decomposition method is a method that divides the original domain into some overlapping subregions whose shapes are regular, hence the problem defined on the original region is translated into the problem defined on the overlapping subregions. Some fast algorithms can be adopted on these subregions, for example the Fast Fourier Transform(FFT), the spectral method. The overlapping domain decomposition method can be used to solve the complicated problems by virtue of parallel machine, efficiently. In the paper, we develop a parallel iterative algorithm based on overlapping Schwarz methods for such a non-Hermitian, non-coercive problem with a complex-valued solution. At the end of the paper, numerical results are given to show the efficiency of the algorithm.In this paper,we consider the following model problem with Robin boundary condition:whereΩ= [0,1]×[0,1], i =(?),k > 0, k denotes the wave number,The exact solution is u{x,y)= DividingΩinto four overlapping subregionsΩ₁,Ω₂,Ω₃ andΩ₄,whereΩ₁= [0,0.6]×[0.4,1],Ω₂=[0.4,1]×[0.4,1],Ω₃ = [0.4,1]×[0,0.6],Ω₄ = [0,0.6]×[0,0.6],u_k is the solution of the original equation on the overlapping subregions that areΩ_k,(k = 1,2,3,4).So we can build the new Schwarz algorithm as follows:Step1. order u₂⁰ - u⁰|_Ω2,u₄⁰ = u⁰|_Ω4,order n:=0.Step2. we will solve the boundary problems onΩ₁ andΩ₃ in parallel.Step3. we will solve the boundary problems onΩ₂ andΩ₄ in parallel. Stcp4. order n:=n+1,go to stcp2.These subproblems are discrete by the finite-difference method.The proposed method can be carried over to the case of more subregions.