Domain Decomposition Algorithm Based On Natural Boundary Reduction Over Unbounded Domain

Many problems of Science and Engineering can be attributed to thesolution problems of partial differential equations, usually the designateddomain of these problems is unbounded, and so the research about solutionalgorithm is particularly important on the unbounded domain. Amongthese, domain decomposition algorithm is prominent, and it’s a new usedwidely, but for solving boundary value problems on the unbounded domain,we cannot get the result that we are satisfied relying on domaindecomposition algorithm because of particularity and difficulty of theexistence of such problems. So, people with the help of the principle ofnatural boundary reduction tried to join the artificial boundary, byintroducing the typical artificial boundary, the original unbounded domaincan be decomposed into a small bounded domain and a typical boundary ofunbounded domain, so in the bounded domain can be solved by thestandard finite element method, and in the unbounded domain can besolved by applying the principle of natural boundary reduction directly. Sothat it would reduce the solving scale and can be used for parallelcomputing.In this paper, a non-overlapping domain decomposition algorithm anda parallel overlapping domain decomposition algorithm which are based onthe circular artificial boundary for solving the two-dimensional exteriorbi-harmonic problems is offered and researched by the principle of naturalboundary reduction. For the non-overlapping domain decompositionalgorithm, the algorithm is constructed and the convergence of its discrete problems is discussed. it is proved that the convergence rate of thealgorithm is independent of the finite element mesh size, with properselection of the relaxation factor it is proved that the convergence rate ofthe algorithm is geometrical, and finally summarized some characteristicsof this algorithm, it shows the this method is very efficient for solving theunbounded domain problems. For the parallel overlapping domaindecomposition algorithm, the algorithm is constructed and the convergenceis discussed, and then gives the discrete form and the processing methodby finite element of this algorithm, and finally summarized someadvantages of this algorithm, it shows that this method is very efficient forsolving the large problems.