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A cylindrical-grid finite-volume (FV) algorithm for the solution of time-harmonic Maxwell's equations in fully three-dimensional (3-D) anisotropic media is presented. To circumvent ill-conditioning and convergence problems, Maxwell's equations are reformulated in terms of potentials in the null space of the curl operator and its complement by applying a Helmholtz decomposition to the electric field. A staggered-grid in cylindrical coordinates is used to better conform to the typical geometries of well-logging problems. The resulting sparse linear system is solved by preconditioned Krylov-subspace methods. FV results are validated in earth formations with anisotropic conductivity against both finite-difference time-domain and numerical mode matching results.