The purpose of a multi-element high lift device is to increase lift dramatically while controlling the stall limit. The fluid flow over a multi-element high lift device has been explored widely both experimentally and numerically at high Reynolds numbers (O(10^6 )). The numerical simulations use turbulence models and hence details of the flow are not yet available. Low Reynolds number (O(10^4 )) flows over high lift devices have not been explored until recently. These lower Reynolds number flows have applications in the development of small aerial vehicles. The present work discusses both two-dimensional and three-dimensional direct numer- ical simulations of fluid flow over a 30P30N three-element high lift system using a high-order spectral element method code, Nek5000, that solves the incompressible Navier-Stokes equations. The intricate geometry of the multi-element device poses a challenge for the high-order spectral element method. We study the complex flow physics in the slat cove region and the wake/shear layer interaction over a 30P30N three-element high lift device. The targeted cases are at Reynolds num- bers based on stowed chord lengths (Rec ) of 8.32 × 10^3 , 1.27 × 10^4 , and 1.83 × 10^4 at angle of attack of 4. A critical interval for Rec has previously been found between 1.27 × 10^4 and 1.38 × 10^4 in experiments. This divides the flow into two types: when Rec is below the critical interval, no roll-up is observed in the slat cove and Görtler vortices dominate the slat wake; however when the Rec is above the critical interval, a roll-up is observed in the slat cove and co-existence of streamwise and spanwise vortices is confirmed in the slat wake. We confirm the presence of the critical interval from the simulations performed at three values of Rec . Lift and drag analysis is provided along with pressure coefficient plots for each element of the multi-element airfoil. Different vortical structures are also identified in the transition of flow from two dimensions to three dimensions. The relevant validation is performed with the available experimental data.