Parallel Stabilized Finite Element Methods Based On The Equal-order Elements For The Stationary Navier-Stokes Equations

The Navier-Stokes equations are typically nonlinear partial differential equa-tions describing the motion of an incompressible viscous Newtonian fluid.It is of significance for people to study and control turbulence.However,we know only a little about the nature of the nonlinear phenomenon,so that it is difficult to seek the exact solution of the Navier-Stokes equations.Consequently,numerical simulations have become an important method to understand the behavior of the solutions.In the modern sciences and engineering computing,the large-scale numerical simula-tions may require massive computing resources that can only be realized by high-performance supercomputers or a cluster of workstations to meet the demands of memory and computing due to the complexity of fluid flow domain.Therefore,it is essential to develop simple and efficient parallel numerical methods with the help of parallel computing technology,in order to quickly realize the large-scale numerical simulations and computations of the Navier-Stokes equations.The main studies of this paper are as follows.For the lowest equal-order P1-P1 finite elements,the third chapter proposes three parallel iterative stabilized finite element algorithms based on a fully overlap-ping domain decomposition technique,where the stabilization term is the pressure projection stabilization based on two local Gauss integrations at the element level.The main idea of the proposed algorithms is that each processor independently com-putes a local stabilized finite element solution in its own subdomain using a locally refined global mesh,so that the algorithms can realize the corresponding parallel computing by slightly modifying the existing Navier-Stokes sequential codes.The algorithms are easy to implement and have low communication complexity.Under some(strong)uniqueness conditions,the stability of three parallel iterative stabi-lized finite element algorithms is analyzed.Using the theoretical tool of local a priori error estimate for stabilized finite element solution,error bounds of the solutions in terms of the velocity and pressure from the proposed algorithms are derived.Finally,some numerical experimentations,compared with the related numerical algorithms,are given to validate the effectiveness of the algorithms.The fourth chapter proposes three parallel stabilized finite element algorithms based on two-grid discretizations.In these algorithms,a global stabilized nonlinear Navier-Stokes problem is firstly solved by an Oseen iterative method on a coarse grid,and then local stabilized and linearized Navier-Stokes problems are indepen-dently solved on overlapping local fine grid in parallel to correct the coarse grid solution,where the stabilization terms both for the coarse and fine grid problems are the pressure projection stabilization based on two local Gauss integrations at the element level.Theoretical and numerical results show that with appropriate s-calings of the algorithmic parameters,the proposed algorithms can yield an optimal convergence rate.Numerical tests show that the parallel Stokes-linearized and sta-bilized algorithm takes less computational time,and the parallel Oseen-linearized and stabilized algorithm is the best one for simulation of large Reynolds number flows or small viscosity coefficient among the three parallel algorithms.For the high order P2-P2 finite elements,the fifth chapter presents three parallel stabilized finite element iterative algorithms based on fully overlapping domain de-composition technique for numerically solving the Navier-Stokes equations,in which the stabilization term is the pressure gradient projection stabilization based on two local Gauss integrations at the element level.Similar to the third chapter,stability and convergence theory of the proposed algorithms are analyzed.Compared with the existing algorithms,numerical tests are also performed to demonstrate the high efficiency of the proposed algorithms.For the high Reynolds number flows,the sixth chapter proposes a two-level stabilized quadratic equal-order finite element variational multiscale algorithm.The algorithm consists of two steps.Firstly,a coarse stabilized solution is obtained by solving a stabilized nonlinear problem on a coarse grid,and then the solution is corrected by solving a stabilized linear problem on a fine grid,where the stabilization terms both for the velocity and pressure are based on two local Gauss integrations at the element level.Under the condition of(?),stability of the presented method is analyzed,and error bounds of the approximate solutions from the proposed method are deduced.The effectiveness and efficiency of the proposed method are demonstrated via some numerical experimentations.