This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions. Figure 1: Example of a two-dimensional non-overlapping domain decomposition method. quasi-optimal convergence properties. Indeed, we will show that the rate of convergence is optimal on the evanescent modes and is significantly improved compared to current techniques for the remaining modes. In the effective convergence, this results in a DDM independent of the wavenumber as well as the mesh discretization. Moreover, we will see how the resulting approximate DtN operator is easy to implement in a basic finite element solver. The paper is organized as follows. In Section 2, we introduce the scattering problem as well as the nonoverlapping DDM. We present in the third Section a non-local square-root operator which approximates the exact DtN transmission operator. Section 4 develops a convergence analysis for this approximate transmission condition on a model problem. Section 5 details the complex Padé approximation of the square-root operator to get a local representation. Section 6 presents the finite element implementation of the resulting DDM. Numerical results on both two-and three-dimensional problems are presented in Section 7.
More recently Boubendir et al. REF presented a quasi-optimal optimized Schwarz method using Padé approximations of the Dirichlet to Neumann (DtN) map.
A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation
J. Comput. Phys.
J. Comput. Phys.
Computer Science Mathematics