High Efficient Numerical Methods For Incompressible Navier-Stokes Equations

Navier-Stokes equations are established for describing the motion of a fluid, which play fundamental roles in fluid mechanics and have wide and important applications in many areas of science and engineering. Due to the high nonlinearity of such equations, it is almost impossible to get the closed form of a solution. Hence, numerical simulation is indispensable in seeking the solution of the Navier-Stokes equations. The goal of this thesis is intended to devise high efficient numerical methods for the incompressible Navier-Stokes equations, discuss convergence rate analysis of the proposed methods and perform a series of numerical results to show the effectiveness of these methods.First of all, some Uzawa-type iterative methods are devised to solve the steady incom-pressible Navier-Stokes equations discretized by mixed element methods. We give the dis-crete analogue of the Uzawa method due to Temam [1]. By developing some novel techniques for convergence rate analysis of an iterative method and using some ideas in [2] technically, we prove that the method converges geometrically with a contraction number independent of the finite element mesh size, even for regular triangulations. Moreover, we emphasize that the previous argument can also apply to the Uzawa method in [1] directly, and as a by-product we can develop convergence rate analysis for this method. The discrete Uzawa method just mentioned is essentially a nonlinear iterative method, and have to solve a system of nonlinear equations at each iteration step. We suggest a linearized iterative method for such a subproblem. As a consequence, we devise an inner and outer iterative method. How-ever, it is often very difficult to set appropriate inner stopping criterion to totally balance computational cost and accuracy, for an iterative method with inner iteration. We luckily find by numerical experiments that one inner iteration is enough to ensure the convergence of the above algorithm. We call such method the modified Uzawa method. By virtue of very careful and technical derivation, we can also prove that this update method still converges geometrically with a contraction number independent of, as the original Uzawa method. A series of numerical experiments are reported to show the computational performance and accuracy of our methods proposed. Next, we prove the Arrow-Hurwicz algorithm for the steady incompressible N-S equa-tions converges geometrically. In the seminal monograph [1], Temam proposed an Arrow-Hurwicz algorithm for solving the steady incompressible Navier-Stokes equations and proved its convergence. However, the convergence rate analysis remains open until now. Us-ing some techniques in [2] combined with a very tricky analysis, we prove that the previous method converges geometrically. This result is important in theory, showing a fundamen-tal property of the Arrow-Hurwicz algorithm. Moreover, it motivates us to devise efficient algorithms for numerically solving the steady incompressible Navier-Stokes equations. We then present the discrete analogue of the Arrow-Hurwicz algorithm. We find when the pa-rameter p in the Arrow-Hurwicz algorithm takes the value v-1, then the algorithm coincides with the modified Uzawa method. p-1can be considered as an adjustable artificial viscosity coefficient, and hence the subproblem of the Arrow-Hurwicz algorithm can be solved more efficiently than that of the modified Uzawa algorithm. We also develop the optimal conver-gence rate analysis of the discrete A-H algorithm and provide a series of numerical results to show the efficiency of this method.At last, we investigate numerical methods for the time-dependent incompressible N-S equations. The gauge-Uzawa method, proposed by Nochetto and Pyo[3], is a kind of the projection method. The method combines both advantages of the gauge method and the Uzawa method. We combine the gauge method and Arrow-Hurwicz method to achieve a method called the gauge Arrow-Hurwicz algorithm. The subproblem of the gauge Arrow-Hurwicz algorithm may be solved more efficiently than the forgoing method. We derive error analysis of the Arrow-Hurwicz algorithm following some ideas in[3].