Finite Element Solution of the Two-dimensional Incompressible Navier-Stokes Equations Using MATLAB

The Navier-Stokes equations are fundamental in fluid mechanics. The finite element method has become a popular method for the solution of the Navier-Stokes equations. In this paper, the Galerkin finite element method was used to solve the Navier-Stokes equations for two-dimensional steady flow of Newtonian and incompressible fluid with no body forces using MATLAB. The method was applied to the lid-driven cavity problem. The eight-noded rectangular element was used for the formulation of element equations. The velocity components were located at all of 8 nodes and the pressure variable is located at 4 corner of the element. From location of velocity components and pressure, it is obvious that this element consists of 16 unknowns for velocities and 4 unknowns for pressure. As a result, the unknown variables for velocities and pressure are 20 per each element. The quadratic interpolation functions represent velocity components while bilinear interpolation function represents pressure. Finite element codes were developed for implementation. The numerical results were compared with benchmark results from the literature.