Optimized Schwarz Methods For The Helmholtz Equation

Helmholtz equationhas been widely applied in physics, engineering and scientific computation. It'ssignificant to research e?cient numerical methods for solving Helmholtz equationin dealing with many physical questions in some fields, such as electromagnetics,acoustics and so on.Domain decomposition methods, developed from Schwarz alternating method(1870), have been widely studied recently, due to the advantages of reducing thelarge-scale problem into smaller ones, reducing the complex problems into simplerones and turning serial computing into parallel computing. After several dozensyears of development, especially with the rapid development of parallel machineand parallel computing technique, corresponding theoretic researches of domaindecomposition methods have made great progress. Its applications have graduallyextended to many fields. In the last decent, one of domain decomposition methods,Schwarz waveform relaxation method, has obtained widespread promotion andapplication.In this paper, we mainly study classical Schwarz methods and its waveformrelaxation variant for Helmholtz equation. Firstly, we analyze the convergenceof classical Schwarz method for Helmholtz equation. It shows that the classicalSchwarz algorithm is not e?ective for Helmholtz equation. Especially, the algo-rithm diverges when the subdomains have no overlap. Nevertheless, the algorithmcan be applied to Helmholtz equation by changing the transmission condition fromDirichlet condition for the classical Schwarz case to general Robin condition. In thepaper, the continuous and discrete optimal transmission conditions for the Schwarzalgorithm without overlap for the Helmholtz equation are studied. These conditionare, however, nonlocal in nature. In this paper, we introduce local approximationsfor both continuous and discrete cases, which may optimize the performance of theSchwarz method. We conclude such algorithm in the class of optimized Schwarzmethods, which belong to waveform relaxation Schwarz algorithm. Moreover, sev-eral kinds of promotions of optimized Schwarz method are presented. Numericalresults illustrate the e?ectiveness of the optimized Schwarz methods we proposed.