Convergence Analysis of the Continuous and Discrete Non-overlapping Double Sweep Domain Decomposition Method Based on PMLs for the Helmholtz Equation

In this paper we will analyze the convergence of the non-overlapping double sweep domain decomposition method (DDM) with transmission conditions based on PMLs for the Helmholtz equation. The main goal is to establish the convergence of the double sweep DDM of both the continuous level problem and the corresponding finite element problem. We show that the double sweep process can be viewed as a contraction mapping of boundary data used for local subdomain problems not only in the continuous level and but also in the discrete level. It turns out that the contraction factor of the contraction mapping of the continuous level problem is given by an exponentially small factor determined by PML strength and PML width, whereas the counterpart of the discrete level problem is governed by the dominant term between the contraction factor similar to that of the continuous level problem and the maximal discrete reflection coefficient resulting from fast decaying evanescent modes. Based on this analysis we prove the convergence of approximate solutions in the H¹-norm. We also analyze how the discrete double sweep DDM depends on the number of subdomains and the PML parameters as the finite element discretization resolves sufficiently the Helmholtz and PML equations. Our theoretical results suggest that the contraction factor for the propagating modes depends linearly on the number of subdomains. To ensure the convergence, it is sufficient to have the PML width growing logarithmically with the number of subdomains. In the end, numerical experiments illustrating the convergence will be presented as well.