Arbitrary high-order C0 tensor product Galerkin finite element methods for the electromagnetic scattering from a large cavity

Abstract

The paper is concerned with the electromagnetic scattering from a large cavity embedded in an infinite ground plane. The electromagnetic cavity problem is described by the Helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and Dirichlet (or Neumann) boundary conditions on the walls of the cavity. A tensor product Galerkin finite element method (FEM) is proposed, in which spaces of C0 piecewise polynomials of degree κ⩾1 are employed. By the fast Fourier transform and the Toeplitz-type structure of the approximation to the nonlocal operator in the nonlocal boundary condition, a fast algorithm is designed for solving the linear system arising from the cavity problem with (vertically) layered media, which requires O(N2logN) operations on an N×N uniform partition. Numerical results for model problems illustrate the efficiency of the fast algorithm and exhibit the expected optimal global convergence rates of the finite element methods. Moreover, our numerical results also show that the high-order approximations are especially effective for problems with large wave numbers.