Complicated partial differential equations (PDEs) frequently appear in the modeling of natural phenomenon. One such example is the system of Navier-Stokes (NS) equations that govern the motion of a fluid in space. Despite its immense importance in science and engineering, the theoretical understanding of the solutions to these equations is incomplete due to the complex turbulent behavior. As the first step towards understanding turbulence, the 3D Navier-Stokes existence and smoothness proof was made one of the seven Millennium Prize problems in mathematics by the Clay Mathematics Institute in May 2000.

Numerical analysis provides a powerful tool to approximate these solutions and to help predict the motion of the underlying fluid flow. In this talk, three standard numerical approaches will be introduced including Finite Difference and Finite Element schemes, as well as a special meshless method termed the Method of Fundamental Solutions. All have been successfully implemented to solve linearized NS equations in a wide range of scales and geometries. Numerical results will be presented to showcase the effectiveness and limitations of these methods.