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A three-dimensional (3D) finite-element formulation for calculating Meissner currents in multiply connected superconductors is presented. The fluxoid quantization condition is ensured as simply as possible. The problem is formulated so that we have to solve two systems of equations by the use of a conjugate gradient algorithm without preconditioning. Meissner currents and magnetic-flux density are numerically evaluated in a superconducting tube and around a vortex. These results are compared with analytical solutions.