The optimal control of fluid flows described by the Navier-Stokes equations requires massive computational resources, which has led researchers to develop reduced-order models, such as those derived from proper orthogonal decomposition (POD), to reduce the computational complexity of the solution process. The object of the thesis is the acceleration of such reduced-order models through the combination of POD reduced-order methods with finite element methods at various discretization levels. Special stabilization methods required for high-order solution of flow problems with dominant convection on coarse meshes lead to numerical data that is incompatible with standard POD methods for reduced-order modeling. We successfully adapt the POD method for such problems by introducing the streamline diffusion POD method (SDPOD). Using the novel SDPOD method, we experiment with multilevel recursive optimization at Reynolds numbers of Re=400 and Re=10,000.