| The topic of this thesis focuses on high order finite/spectral element methods for solving unsteady incompressible viscous Navier-Stokes equations. The gauge formulation of the Navier-Stokes equations was proposed by Weinan E and Jian-Guo Liu recently [10]. The main advantage is that the gauge variable is nonphysical, so we have the freedom to assign boundary conditions.;In many cases, the low order methods are not able to catch up the complicated flow structures, to achieve good accuracy for the pressure term, or the divergence free condition. In this thesis, we focus on developing high order finite/spectral element methods for solving Navier-Stokes equations.;Based on the gauge formulation, several high order finite/spectral element methods based on the gauge formulation are developed. For the time stepping procedures, backward Euler and Crank-Nicholson methods are used. The numerical experiments show clean high order accuracy. Some high order time-stepping procedures such as the backward differentiation methods or the explicit forth-order Runge-Kutta method have also been tried. In all these methods, the computations of the gauge variable and the auxiliary field are decoupled, with the nonlinear term treated explicitly. All the computations are reduced to solving heat equations and Poisson equations, so they are very efficient.;Based on the vorticity-stream function formulation of the Navier-Stokes equations, Jian-Guo Liu and Weinan E [22] proposed an efficient time-stepping procedure recently so that the computations of the vorticity function and the stream function are decoupled, with the nonlinear term treated explicitly. The second part of the thesis is the implementation of the procedure by using high order finite/spectral element methods for space discretization. The numerical experiments are presented including clean high order accuracy and the simulations of the canonical driven cavity flows.;Preconditioned conjugated gradient (CG) methods are used to solving the resulting linear systems. We have studied the convergence property of the CG method when applied to symmetric positive semi-definite systems.;To exploit the sparsity structures of the stiffness matrix and mass matrix, the local assembly technologies are used to evaluate the nonlinear terms, various right hand sides, the gradients and the matrix-vector product. |
