Summary form only given. We have constructed mimetic finite difference methods for both the TE and TM modes for 2-D Maxwell,s curl equations and equations of magnetic diffusion with discontinuous coefficients on nonorthogonal, nonsmooth grids. The discrete operators were derived using the discrete vector and tensor analysis to satisfy discrete analogs of the main theorems of vector analysis. Because the finite difference methods satisfy these theorems, they do not have spurious solutions and the "divergence-free" conditions for Maxwell's equations are automatically satisfied. The tangential components of the electric field and the normal components of magnetic flux used in the FDM are continuous even across discontinuities. This choice guarantees that problems with strongly discontinuous coefficients are treated properly. Furthermore on rectangular grids the method reduces to the analytically correct averaging for discontinuous coefficients. We verify that the convergence rate was between first and second order on the arbitrary quadrilateral grids and demonstrate robustness of the method in numerical examples.