Discontinuous Finite Element Methods For The Incompressible Navier (Navier-Stokes) Equations

The discontinuous Galerkin method was introduced by Reed and Hill [31], and extended by Cockburn and Shu [15, 16, 17] to conservation law and system of conservation laws,respectively. Due to localizability of the discontinuous Galerkin method, it is easy to construct higher order element to obtain higher order accuracy and to derive highly parallel algorithms. Because of these advantages, the discontinuous Galerkin method has become a very active area of research. Discontinuous Galerkin method have been used to solve hyperbolic and elliptic equations [1, 5, 10] and Stokes equations [7, 18, 34] by many researchers. The purpose of the thesis is to discuss how to solve the incompressible Stokes(Navier-Stokes)equations by DG methods.This thesis is composed of two parts.In the first part, we introduce the discontinuous Galerkin method for the incompressible Stokes equations; In the second part, according the specific feature of the Navier-Stokes equations, we utilize the Characteristics-based method and combine it with mixed discontinuous finite element methods to solve the nonstationary Navier-Stokes equations.In chapter two,we derive a new discontinuous finite element formulation for the Stokes equations, based on the general discontinuous Galerkin methods and the pressure gradient local projection method of the Stokes problems [11]. This new formulation is stable any combination of discrete discontinuous velocity and pressure spaces, when polynomials of degree l are used for each component of velocity and polynomials of m for the pressure, for any l ≥ 1 and m ≥ 0. Optimal error estimates for the approximation of both velocity and pressure in L~2 norm...