Stabilized Methods For The Stationary Incompressible Flow Problems

The stationary incompressible flow can be seen as a flow in steady, which de-scribes the motion law of some fluids. For example, the movement of the atmosphere,the transport in the ocean, the flow in the turbomachinery and so on. Particularly,it is very important to help people to understand and control turbulent. The maingoverning equations are the incompressible Navier–Stokes equations. Because it isnot very easy to realize the essence of the nonlinear phenomena, the numerical meth-ods have played an important role. However, numerical simulation of Navier–Stokesequations has a great difculty, i.e., the contradiction between huge problem sizeand very limited computing ability. Hence, constructing and studying an algorithmwith good stability and convergence are very important. In this thesis, based on thelocal Gauss integration, several stabilized methods for the stationary incompressibleflow are studied as follows:1,Based on the local Gauss integration, we propose a novel defect-correctionmethod for the stationary Navier–Stokes equations. The method uses a bilinear termto replace an artificial viscosity stabilized term (i.e. adding to the bilinear form thediference between an exact Gaussian quadrature rule for quadratic polynomialsand an exact Gaussian quadrature rule for linear polynomials to ofset the inf-supcondition). Diferent from the common defect-correction method, it is independentof mesh size. Besides, from the numerical experiment, we can see that the results ofour method are much better than those of the common defect-correction method.2,The two-level quadratic equal-order stabilized finite element method for thestationary Navier–Stokes equations based on the local Gauss integration is consid-ered. The method includes three corrections: Stokes correction, Newton correctionand Oseen correction. The theoretical analysis and numerical results confirm thatthe Stokes and Newton corrections are useful for large viscosity and the Oseen cor-rection is the best method for small viscosity. Moreover, two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokesequations based on the local Gauss integration are considered. The methods combinethe defect-correction method and the two-level strategy with the locally stabilizedmethod. It includes three algorithms: m defect steps by Oseen iteration and oncecorrection by Oseen iteration (m-Oseen-1-Oseen); m defect steps by Oseen itera-tion and once correction by Simple iteration (m-Oseen-1-Simple); m defect steps byOseen iteration and once correction by Newton iteration (m-Oseen-1-Newton).3,Applying an L2-projection, we present the superconvergence results for thestationary Navier–Stokes equations by the stabilized nonconforming finite elementmethod and the stabilized finite volume method. The basic idea is to project theapproximate solution to another finite dimensional space on a diferent, but coarsermesh. The diference in the two mesh sizes can be used to achieve a superconvergenceafter the post-processing procedure. Numerical results are shown to support thedeveloped theory analysis.4,A two-level stabilized finite element method for the Stokes eigenvalue prob-lem based on the local Gauss integration is considered. It provides an approximatesolution with the convergence rate of same order as the usual stabilized finite ele-ment solution. Hence, our method can save a large amount of computational time.Numerical tests confirm the theoretical results of the presented method. Moreover,several stabilized finite element methods for the Stokes eigenvalue problem basedon the lowest equal-order finite element pair are numerically investigated. They arepenalty, regular, multiscale enrichment, local Gauss integration, and nonconforminglocal Gauss integration method. Comparisons between them are carried out, whichshow that the nonconforming local Gauss integration method has good stability,efciency and accuracy properties and it is a favorite method among these methodsfor the Stokes eigenvalue problem.