Evan Johnson

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

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