An accurate conformal Fourier transform method for 2D discontinuous functions

Fourier transform of discontinuous functions are often encountered in computational electromagnetics. A highly accurate, fast conformal Fourier transform (CFT) algorithm is proposed to evaluate the finite Fourier transform of 2D discontinuous functions. A curved triangular mesh combined with curvilinear coordinate transformation is adopted to flexibly model an arbitrary shape of the discontinuity boundary. This enables us to take full advantages of high order interpolation and Gaussian quadrature methods to achieve highly accurate Fourier integration results with a low sampling density and small computation time. The complexity of the proposed algorithm is similar to the traditional 2D fast Fourier transform algorithm, but with orders of magnitude higher accuracy. Numerical examples illustrate the excellent performance of the proposed CFT method.