The Discontinuous Galerkin Methods For Solving Navier-Stokes And Fractional Equations

In this thesis, we consider two kinds of di?erential equations, namely the timedependent incompressible Navier-Stokes equations and space-fractional RiemannLiouville equations in two dimensions(2D).For Navier-Stokes equations, ?rstly, we introduce an auxiliary variable to split the di?usion operator and make the high-order equation into one order system, then we reduce the di?culty of the high-order problem. Secondly, making the variational formulas, carefully choosing numerical ?uxes and adding penalty terms, we propose a stable and symmetric local discontinuous Galerkin(LDG) scheme. So we ?nish the space discrete for the problem, but we do not discretize the convection term. As we know, generally there are two di?culties for solving the Navier-Stokes equations,i.e., how to deal with the nonlinear convection term and how to tackle the pressure function in Navier-Stokes equations. Since the characteristic method have the advantage in solving the advection-dominated di?usion equations, then we consider the characteristic method to discretize the time derivative term and nonlinear convection term together. Under it we solve the nonlinear di?culty, and obtain a stable and symmetric characteristic local discontinuous Galerkin(CLDG) scheme. When performing the stability analysis, we ?nd that the characteristic method makes the theoretical deduction easier. Especially, applying the symmetry of the scheme, some complicated operators have been eliminated. Because the velocity function u and the pressure function p do not have the closed relation in the Navier-Stokes equations, then after getting the error estimate of velocity, it is not easy to get the estimate of pressure. In order to tackle this di?culty, we use the continuous inf-sup condition for velocity and pressure to make a connection for velocity and pressure.In fact, in discontinuous ?nite element space we use the discrete inf-sup condition,but in this thesis, we combine the continuous inf-sup condition and discrete ?nite element space, and successfully obtain the estimate for pressure, which is one of the highlights in Chapter 3. Finally, we give four di?erent examples to verify theoretical results, and ?nd that the numerical results reach the desired results and even much better.For the space-fractional Riemann-Liouville equations in 2D, ?rstly, we introduce two auxiliary variables to split the Riemann-Liouville derivative. Since the RiemannLiouville derivative has the singularity, Riemann-Liouville integral does not have the sigularity, then we use one auxiliary variable to substitute the grad term, another one to substitute the Riemann-Liouville integral. So we split the Riemann-Liouville derivative and make the high-order equation into the one order system. Secondly,making the variational formulas, carefully choosing numerical ?uxes and adding the penalty terms, we obtain a hybridized discontinuous Galerkin(HDG) scheme, then we accomplish the semi-discrete process. Finally we give three ways to perform the time discretization, i.e., for the Riemann-Liouville di?usion problems, we use the general ?nite di?erence method to discretize the time derivative, and we omit the discrete process because of the simpleness. For the Riemann-Liouville convection di?usion problems, we use one order and two order characteristic method to discretize the time derivative and convection term together. And we give the corresponding fully discrete formulas, stability analysis, error estimates. When studying the space fractional Riemann-Liouville equations, we ?nd it is not easy to ?nd an e?cient method to perform the theoretical analysis, especially for the error estimates. From this point, we do not think of an e?cient analytical tool, but consider the way of designing the numerical formula. Based on this idea, we obtain the di?erent numerical formula, extend the one dimensional case into two dimensional case, and get the better numerical results. In the last section of the chapter 4, we use three numerical examples to verify the Riemann-Liouville equations and the one order and the two order HDG scheme, respectively.Above all, in this paper we successfully combine the discontinuous Galerkin method and the characteristic method to solve the Navier-Stokes equations and space-fractional di?erential equations in 2D. Note that the numerical results are consistent with theoretical results. Then, these schemes can be further studied and applied into other problems.