A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions.  We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local $hp-$refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.