The Nonoverlapping DDM For 2-D Helmholtz Problem

With the development of science and technology and thorough research ofvarious problem, the people discover that the transmission problems inwaveguide of the electromagnetic field, the elasticity waveguide problems andthe diffraction problems in water wave of the ocean project are all based onHelmholtz equation. Therefore, how to solve the Helmholtz equation rapidly andaccurately is becoming more and more important. The domain decompositionmethod (DDM) is the method of dividing the original domain into somesubregions whose shapes are regular, hence the problem solving on the originalproblem is turned to the problems solving on subregions. The fast algorithm canbe choose on the regular subregions such as the fast Fourier transform ( FFT), thespectral method etc.. The DDM is used to solve the complicated problem.Considering 2-D Helmholtz problem of DirichletDividing ? into two nonoverlapping subregions Ω₁ and Ω₂,whereΩ₁ =[0,1]×[0,1], Ω₂ =[1,2]×[0,1], Γ ={( x ,y)|y∈[0,1],x=1}.The above-mentioned problem equal to solve the Dirichlet problem on ? 1?????=??Γ=Γ?+=?,(\).,(),0,(),1111121uguukunnnnnλ (2)and the Neumann problem on ? 2???????=??Γ??=??Γ?+=?,(\).,(),0,(),2221222222ugnunuukunnnnn(3)The finite-difference method is presented for computing the two subproblems.Solving(2)equal to solveAu = b,whereA = ???????? ?OB I ??OB II ??OB II ?BI????????,I is an identity matrix in R ( M?1 )(M?1), B is a three-diagonal matrix inR ( M?1 )(M?1).B = ???????? 4 ??Oh 12 k24???Oh 112 k24???Oh 112 k24??h12k2????????,Tu = (u 1, 1,u2,1,L ,uM ?1,1,u1,2,u2,2,L,uM?1,2,L,u1,M?1,u2,M?1,L,uM?1,M?1),TMmMMbmMmMmMM0,,0,)0,,0,,sin((1)/),(sin(/),sin(2/),,sin((2)/),121??+=?λλεπλεπεπεπLLLLSolving(3)equal to solve A ′ u′=b′,whereA′ =????????O BC′ OBCC′ OBCC′ BC′????????,B′ ,C are matrices in R M× M. I is an identity matrix in R ( M?1 )(M?1).B ′ =???????? ?L100 1 4???Lh0 112 k24??Lh0012k2?L0001LLLLL ?L00014?Lh0002k2????????,C = ??? 00 ?0I???,,,,),(,,,,,,,,,11,121,1,11,121,1,21,221,2TMMMMMMMMMMMMvvvvvvvvvv?+???=+?+?LL LL,0,,0).,0,,0,sin((22)/),sin((21)/),(,sin((1)/),sin((2)/),,11,121,211,1TMMMMMuumMMmMMbumMMmMMLLLL??????+?+??′=?+++λλεπεπλεπεπThough the DDM on two subregions can't completely reflect the superiority ofthis method, it is a research foundation on many subregions. Therefore carryingon this research is still necessary and very worthy.