Highly storage efficient choices of boundary meshes are devised for hybrid finite element - boundary element analysis in 3-dimensions, and a computationally-efficient algorithm to solve the resulting matrix equations is developed. The boundary mesh makes use of cylindrical symmetry to produce block circulant boundary element matrices with proper choices of basis functions, and results in up to 64-fold reduction in matrix storage. Also the block circulant nature of the boundary element matrices provides a numerically efficient algorithm for solution of the coupled matrix equations. To demonstrate its efficiency this hybrid scheme is applied to a three-dimensional magneto-static field problem, and proves storage and computational efficiency. With this efficiency the hybrid scheme becomes more advantageous than pure finite element method with approximate boundary conditions, and allows to solve large 3-dimensional open boundary field problems in small workstations.