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Highly storage-efficient choices of boundary meshes are devised for hybrid finite element-boundary element analysis in three 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. 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 magnetostatic field problem, proving storage and computational efficiency. This hybrid scheme is more advantageous than the pure finite-element method with approximate boundary conditions. and allows large three-dimensional open boundary field problems to be solved in small workstations.