Spectral element based numerical solvers are developed to simulate electrokinetically driven flows for micro-fluidic applications. Based on these numerical solvers, basic phenomena and devices for electrokinetic applications in micro and nano flows are systematically studied. As a first application, flow and species transport control in a grooved micro-channel using local electrokinetic forces are studied. Locally applied electric fields, zeta potential patterned grooved surfaces, and geometry are manipulated to control mixed electroosmotic/pressure driven flow in the grooved micro-channel. The controlled flow pattern enables entrapment and release of prescribed amounts of scalar species in the grooves. As another application, hydrodynamic/ electrokinetic focusing in a micro-channel is studied. External electric field, flow rate of pressure driven flow, and geometry in the micro-channel are manipulated to obtain the focusing point, which led to determination of the electrophoretic mobility and (relative) concentration of dilute mixtures. This technique can be used to identify and detect species in dilute mixtures.