This paper is concerned with the Cauchy problem for the Helmholtz equation. Recently, some new works asked the convergence of the well-known alternating iterative method. Our main result is to propose a new alternating algorithm based on relaxation technique. In contrast to the existing results, the proposed algorithm is simple to implement, converges for all choice of wave number, and it can be used as an acceleration of convergence in the case where the classical alternating algorithm converges. We present theoretical results of the convergence of our algorithm. The numerical results obtained using our relaxed algorithm and the finite element approximation show the numerical stability, consistency and convergence of this algorithm. This confirms the efficiency of the proposed method.

中文翻译：

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与亥姆霍兹方程相关的柯西问题的有效松弛交替过程

本文关注亥姆霍兹方程的柯西问题。最近，一些新作品询问了著名的交替迭代方法的收敛性。我们的主要结果是提出了一种新的基于松弛技术的交替算法。与现有结果相比，该算法实现简单，对所有波数选择收敛，可作为经典交替算法收敛情况下的收敛加速器。我们提出了算法收敛的理论结果。使用我们的松弛算法和有限元近似获得的数值结果显示了该算法的数值稳定性、一致性和收敛性。这证实了所提出方法的有效性。