An efficient method for multiple-scale problems is presented, which extends the authors' modified Temam scheme on the staggered mesh [5] to nonuniform staggered mesh. The nonuniform mesh is generated by a continuous coordinate transformation without singlarity according to boundary layer thickness so as to ensure accuracy. The constraint condition, i.e. continuity equation, is realized by the pressure correction projection method; the related Poisson equation on nonuniform mesh is solved by the generalized cyclic reduction program from FISHPACK. Numerical simulation of driven flow in a square cavity was done for steady-state solutions for Re=100, 400, 1000, 5000. These solutions accurately match those given in literature; for high Reynold numbers, tertiary vortices near the corners in the cavity are resolved. These results shows that with suitable nonuniform mesh, the above method remarkably improves computational efficiency.