New Algorithms In Domain Decomposition Without Intersections

Domain decomposition method which has developed recently is a new technique toget numerical solution of PDE.It has many adventages such as flexibility,highlyparallel,and suitable for problems of big scale.In this dissertation the author suggeststhree new methods in domain decomposition without intersections.First,the difference algorithm is suggested.In fact,it is a simplified D-Nalternatemethod.Using the algorithm we can solve the Dirichlet boundary value problems andthe Neumann boundary value problems of Poisson equation at the same time.Themethod has been simplified but the different boundary value problems can be solvedonce.Here we take Poisson equation for instance.The idea of the difference algorithmis not confined to such kind of equation.Also the relaxation parameterθnwhich isequivalent to that in Richardson iteration，being optimum，and the true valueuaregot.The numerical example shows that the computing result is concerned with thetheory of the algorithm.Second, for the outside problem of Poisson equation the Schwarz alternating methodwhich has square convergence rate is proposed.When we get the numerical solutions ofPDE,we generally refine mesh to improve the accuracy.But it increase the amount ofwork greatly.Here using the discrete averages of the functions in Sobolev space forpretreatment,then the iteration is carried out.The new method has square convergencerate in comparison with the priginal one.The numerical example in typical region andthe illustrations show that the new method has finer results.Finally, to select the linear operator on the transmission boundary conditionadaptively in the optimal Schwarz alternating method,we introduce the operatorrelaxation parameters.and the convergence rate can be improved if the relaxationparameters are choosen properly.The results are got under linear and nonlinearconditions.A numerical example shows the same results.