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A finite-volume (FV) technique is introduced to model directional resistivity logging sensors that employ tilted-coil antennas in 3-D anisotropic earth formations. The algorithm is based on a coupled scalar-vector potential formulation on a 3-D cylindrical staggered grid. The choice of discretization enforces the correct continuity equations across material boundaries, avoids spurious modes, and leads to a better conditioned system matrix. In order to model situations where the sensor tool axis does not coincide with the borehole axis (eccentered tools), a locally conformal FV scheme is also implemented. The proposed FV formulation is validated against pseudoanalytical solutions (in cases where the latter is applicable), showing very good agreement, and utilized to investigate the effects of the drill collar (borehole), anisotropy, and azimuthal rotations on the response of directional resistivity sensors.