Domain Decomposition Algorithm For The Problem Of The Plane Elasticity Equation On Unbounded Domain

With the development of parallel technology, domain decomposition algorithmgrew rapidly. Domain decomposition algorithm, i.e. decompose the computationaldomain Ω into mvers subregions,Ω （?）=Ω₁=.The solving of the original1problem transform into solving the problems on the subregions. For solving theproblem on unbounded domain, we usually use the coupling method of the finiteelement and boundary reduction, or use an appropriate artificial boundary and theapproximate boundary condition. Saying, we decompose the infinite domain Ωinto a finite domainΩ₁and a typical infinite domainΩ₂, and solve the problemonΩ₁andΩ₂alternatively. A small scale problem inΩ₁is solved by the finiteelement method, and only simple computation on the typical boundary ofΩ₂isneeded by natural boundary element method. This method can decrease the scale ofcomputation and also can apply parallel computation.In this paper, we mainly take plane elasticity equation as an example, to studyoverlapping domain decomposition methods and non-overlapping domaindecomposition methods based on the finite element method and the theory ofnatural boundary reduction.For overlapping domain decomposition methods, the domain unboundedoutside the boundary is resolved by introducing artificial boundaries. Projectionoperators are defined by finite element theory, and its geometric convergence isproved in the sense ofiV（?）_V（Ω）by using the projection theory.For non-overlapping domain decomposition methods, take the mixed boundaryvalue problem of plane elasticity equation as an example, continuous D-Nalgorithm and over domain outside circle its convergence analysis are given. Wediscuss the convergence of discrete form iteration, and prove the convergence rateis independent of the finite element mesh size. With proper selection of therelaxation factor it is proved that the convergence rate of the algorithm isgeometrical.