Finite Element Methods For The Two-Dimensional Incompressible Navier-Stokes Equations

The Navier-Stokes equations(NSE) are the typical nonlinear equations. It is important tostudy and control turbulence. Nowadays, however, we know only a little about the natureof the nonlinear NS problem. Therefore, numerical simulations and experimentation withreal situation have become an important approach to know and reconstruct the world.Major di?culties consist in large Reynold problem, incompressible condition, unstructuredmesh, inf-sup condition and nonlinear problem as for directly numerical simulation of theNSE. Thus, it is crucial to construct and studybetter algorithms with better stability andconvergence and feasible parallel computation. In order to overcome the above di?culties,we study several algorithms for the incompressible NSE.The third chapter constructs the new stabilized finite element method based on local Gaussintegration for the incompressible ?ows approximated by the lowest equal-order finite ele-ments. This new stabilized finite element method has some prominent features: parameter-free, no higher-order derivatives or edge-based data structures, and stabilization completelylocal at the element level. As for the unstructured mesh, the numerical results of the newstabilized finite element method indicate the same performance as the standard finite el-ement method with the Taylor-Hood element. Here, we study the theoretic results andnumerical experiments for the new stabilized conforming and nonconforming finite elementmethod as well as the stabilized finite volume element method.The fourth chapter implements four finite element methods: the new stabilized finite ele-ment method based on local Gauss integration, two-level stabilized finite element method,superconvergence by coarsening local L2 projection, and Euler time-space iterative finiteelement method. Firstly, we extend the stabilized finite element method based on Gaussintegration to the stationary NSE. As for the stationary NSE, we analyze the results under the classical and non-singularity conditions. The results demonstrate that the stabilizedfinite element method is e?cient all the same for the nonlinear problem. Secondly, wedesign the two-level and multi-level stabilized finite element method. It achieves the sameperformance as the standard Galerkin method, and what is more, it can save much com-putational time. Thirdly, we present superconvergence by coarse local L2 projection. It is?exible to perform for the conforming finit element method, nonconforming finite elementmethod, and discontinous Galerkin finite element method. It does not dwell on the super-convergence point but superconvergenc domain. The postprocessing mesh does not requireuniform regular mesh but regular mesh and thus guarantees that computation can carry onadaptive mesh. Function space can replace the finite element space as to postprocessing.Also, it is independent of problem, which is studied, and the inf-sup condition. Finally,we implement Euler time-space iterative finite element method. It can quickly solve thestationary Navier-Stokes equations with some small viscous, whereas three classical spatialiterative methods can only solve the stationary Navier-Stokes equations with some largeviscous.The fifth chapter discusses some e?cient algorithms for the non-stationary Navier-Stokeequations. Firstly, we obtain the optimal spatial discretization results for the transientNSE under H2 smooth regularity assumptions on the initial solution. Through a series ofnumerical experiments, the new stabilized finite element method has better stability andaccuracy results on both the uniformly and the unstructured mesh than other classicalstabilized finite element method. Finally, we study the numerical implementation of thetwo-order full discretization Crank-Nicolson/Adams-Bashforth scheme proposed by Profes-sor Yinnian He for the non-stationary two-dimensional NSE. The theoretic and numericalresults demonstrate that the scheme can almost obtain the same stability as the Euler im-plicit scheme and has the same performance as the Crank-Nicolson extrapolation scheme interms of convergence. Moreover, the method only stably solves Stokes equations by three-level time iterative, whereas the Euler implicit scheme has to perform on the nonlinearproblem and the Crank-Nicolson extrapolation scheme leads to numerical oscillations inproblems with rough initial data or boundary conditions.