A Numerical Method for the Navier Stokes Equations in a Velocity-Vorticity Form

In physics, the Navier-Stokes equations (NSE) describe Newtonian fluid flows. Instead of focusing on the standard NSE with velocity and pressure as primary variables, we are interested in the Navier Stokes equations in a velocity-vorticity form (NSE-VV). In NSE-VV, vorticity is included in the system as an independent variable so that it can be computed directly. The main contribution of this thesis is proposing a numerical method for solving NSE-VV, which includes an efficient solver, an adaptive mesh refinement strategy and error analysis. A nested solver plays a fundamental role in the numerical method of NSE-VV. Similar to how it is typically done for the NSE, we use Newton’s iteration as the non-linear solver by which a linear PDE system of Newton update can be generated. The finite element method is applied to the system of Newton update and a linear system can be derived from the finite element discretization. Different from the standard NSE, the coupled linear system of NSE-VV is characterized by a 3 × 3 block system matrix. In order to remove the dependence of the spectral distribution on the mesh size, we design an augmented Lagrangian type preconditioner with respect to the special block structure. As a result of the preconditioning method, the number of iterations in solving the preconditioned linear system does not depend on the number of degrees of freedom which increases with mesh refinement. In addition, an operational strategy is proposed aiming to improve computing efficiency. Global mesh refinement is widely used to check convergence, however, it is neither necessary nor practical if a flow contains singularities, or the mesh contains a large number of degrees of freedom in realistic problems. In practice, we would like to obtain accurate results with as cheap computation as possible. Adaptive mesh refinement aims to refine the critical region where the local error is large, instead of the whole domain, such that to reduce the number of cells. A posteriori error estimator is developed based on the residual of the whole system of NSE-VV and only depends on the discrete solution and the problem data and mesh information. As a result, an adaptive mesh refinement strategy is designed based on the posteriori error estimator. A proof of well-posedness of the finite element discretization is derived as the theoretical support of the numerical method. Based on the result of well-posedness, we discuss the convergence of NSE-VV. As a conclusion, the overall convergence of NSE-VV is suboptimal if using Taylor-Hood elements. The corresponding program is developed in parallel with MPI C++ for numerical experiments. The numerical results are computed on the Palmetto Cluster (the high-performance computing system of Clemson University). The program for numerical simulations is based on external libraries, such as deal.II for parallel finite element and Trilinos for parallel linear algebra