The Source Transfer Domain Decomposition Method For The Helmholtz Equation With Large Wave Number Under The Discretiza- Tion Of The Continuous Interior Penalty Finite Element Method

Design and Theoretical analysis of high-efficient algorithms for the scattering prob-lems with large wave number is a famous public problem. Especially because of the "pollution effect", to solve scattering problems with large wive number always needs the solution of large linear algebra equations. Previously one of the algorithms de-signed by Z.Chen and X.Xiang calling the Source Transfer Domain Decomposition Method(STDDM) which is inspired by the "Sweeping" preconditioner developed by B. ENGQUIST and L. Ying.emerged. Just as the "Sweeping" preconditioner, the method nearly got the perfect result in dealing with the Helmholtz equation with large wave number under the discretization of finite element and finite difference method. On the other hand, one discretization method calling the Continuous Interior Penalty Finite Element Method(CIPFEM) which was developed and analyzed by H.Wu L.Zhu and Y.Du have better stability and can enormously reduce the effect of the pollution item.In this paper, we got a new kind of discretization method through discreting the subdomain problems of the STDDM by CIPFEM which is denoted by CIP-STDDM, the calculating cost of the method is O(NM) in which N stands for the number of the sub-domains in the decomposition of the calculating domain, while M is the cal-culating cost of the problem discreted by the CIPFEM in one subdomain. It’s shown in the numerical experiments that the numerical solution obtained by the CIP-STDDM is almost as accurate as the one got by the CIPFEM, however the calculating cost of the the previous one is much smaller which makes it more appropriate for large scale calculation. Besides, we also try to use the stiff matrix of the CIP-STDDM as a pre-conditioner for the CIPFEM to construct a preconditioned GMRES method for solving the CIPFEM problems. Each cost in one iteration of the GMRES method is O(NM). It’s also testified in the numerical experiments that the upper bound of the rate of con-vergence in this GMRES method is independent of the wave number k, the degree of freedom of CIPFEM and the number of subdomains, in conclusion,the algorithm can deal with the CIPFEM equations robustly and efficiently.