A Kind Of Additive Schwarz Procedures Based On Robin Transmission Condition

Schwarz alternating algorithm, regarded as an earliest domain decompositionmethod, was proposed by German mathematician H.A.Schwarz in 1870. However,domain decomposition technique has not been widely applied in large-scale sci-entific and engineering computation until 1980's. Because domain decompositionmethods can decompose large-scale problems into small ones, complex boundaryvalue problems into simple ones, serial problems into parallel ones, many peopleespecially the calculate mathematicians pay attention to this field due to the inven-tion and popularization of parallel machines and networks. Since 1987, the inter-national conference on domain decomposition methods was held annually. Scholarsand researchers in numerical analysis and engineering, from the United States's,the Soviet Union's, France's, Italy's, China's and so on, compete to participate inthis active research.This thesis mainly illustrates an additive Schwarz procedure with non-overlap-ping and overlapping domain decompositon methods based on Robin transmissioncondition for Neumann model problem. The analysis of the convergence of theiterations for both the di?erential form and the mixed finite elemenet approxima-tion form are presented. In the methods, the variable parameter cycle proposed byDouglas and Huang is used. The related results have been extended to the multi-domains problem and the three dimensional case. Finally, we illustrate the relatednumerical results of the additive Schwarz algorithms with two nonoverlapping oroverlapping subdomains.Firstly, we present an introduction to domain decomposition methods, andillustrate the Robin transmission conditions as well as the model problem of theNeumann kind.Secondly, we describe an additive Schwarz procedure with two non- overlap-ping subdomains for the Neumann model problem. The convergence analysis ofthe iteration, for both the di?erential case and the mixed finite element approxi-mation, is presented. And then, we describe an additive Schwarz procedure withtwo overlapping subdomains for the Neumann model problem. The convergenceanalysis of the iteration, for both the di?erential case and the mixed finite elementapproximation, is presented.Furthermore, we extend the two subdomains into multi-domains, and presentan additive Schwarz procedure with multi-domains. We also discuss an additive...