## A fast algorithm for the electromagnetic scattering from a large cavity.(English)Zbl 1089.78024

Summary: A fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane. The medium inside the cavity is assumed to be (vertically) layered. By introducing a transparent (artificial) boundary condition, the problem in the open cavity is reduced to a bounded domain problem. A simple finite difference method is then applied to solve the model Helmholtz equation. The fast algorithm is designed for solving the resulting discrete system in terms of the discrete Fourier transform in the horizontal direction, a Gaussian elimination in the vertical direction, and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system. The existence and uniqueness of the finite difference solution are established for arbitrary wave numbers. Our numerical experiments for large numbers of mesh points, up to 16 million unknowns, and for large wave numbers, e.g., between 100 and 200 wavelengths, show that the algorithm is extremely efficient. The cost for calculating the radar cross section, which is of significant interest in practice, is $$O(M^{2}$$) for an $$M \times M$$ mesh. The proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem.

### MSC:

 78M20 Finite difference methods applied to problems in optics and electromagnetic theory 78A45 Diffraction, scattering 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 65T50 Numerical methods for discrete and fast Fourier transforms 65F10 Iterative numerical methods for linear systems 65F05 Direct numerical methods for linear systems and matrix inversion
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