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We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of twocoupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction-diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.