The investigator develops a class of efficient and accurate numerical methods for the unsteady viscous incompressible Navier-Stokes equations (NSE) based a new unconstrained formulation of NSE with fully dissipation in contrast to the traditional formulation where the Stokes operator is dissipative only in the divergence free fields. This class of NSE solver is unconditionally stable with explicit treatment of both pressure and convection terms. Moreover, in this class of finite element methods, there is no requirement of the so called inf-sup condition. The cost of solving NSE is greatly reduced to solving a standard heat equation and a standard Poisson equation at each time step for general three dimensional fluid problems. The simplicity of the method also enable the PI to develop a class of numerical methods for complex fluids such as magneto-hydrodynamics, liquid crystal polymers, geodynamo, climate modeling, and large eddy turbulence simulations;
Computational Fluid Dynamics has grown from a mathematical curiosity to become an essential tool in almost every branch of fluid dynamics, from aerospace propulsion to weather prediction and has received extensive attention throughout the international community since the advent of the digital computer. The accuracy and efficiency of the proposed schemes will allow us to simulate general three-dimensional time-dependent flows with a reasonable turn-over time. It is expected that the proposed fast algorithms will become an important tool for many scientists and engineers in numerous scientific and industrial applications of current interest.
