Two-level Finite Element Methods For The Steady Natural Convection Problem And Stationary In Compressible Magnetohydrodynamics

Because people have limited understanding for the essence of the nonlinear phenomenon,numerical simulation becomes a very important research measure.But there is a big difficulty in direct numerical simulation of nonlinear partial differential equation,and the difficulty is the contradiction between the huge computational scale,long-time integration and limited computing resource.Thus it is extremely important to structure and research high-effective and low-loss algorithm which has long-time stability.The article takes the natural convection problem and incompressible magnetohydrodynamics(MHD)equations as research objects,and structure two-level finite element algorithms for incompressible fluid.Two-level method is an efficient numerical discretization scheme for solving nonlinear partial differential equation.Applied to the finite element,it can simplify solving complicated nonlinear problem into two steps: The main idea is to firstly find an initial approximation on a coarse mesh;and then to solve a relatively simple linear problem by using the coarse mesh solution on a fine mesh.Thus it is a good strategy to decrease the computational cost.In numerical calculation,iterative method becomes a indispensable method to solve complex nonlinear problem.Iterative method is the repeated iterations that start from an initial estimate and then find a series of approximate solution for the problem.For problem with small viscous coefficients(is high reynolds number),the standard Galerkin finite element method may appear numerical oscillation,which limits development of finite element method in computational fluid dynamics.But stabilization method makes this problem solvable.So subgrid stabilized method will be subjected to a study of this scope.In addition,for complex fluids the pressure and velocity is coupled,which makes solving this problem difficult.But penalty method can well dispose it.So the penalty method is included in the research of this scope,too.Now,thanks to above many effective method,we construct a series of new algorithms for incompressible fluid,as follows:In Chapter 3,aimed at the natural convection problem,based on Oseen iteration and subgrid stabilized method we structure new two-level method: coupled two-level subgrid stabilized Oseen iterative method and decoupled two-level subgrid stabilized Oseen iterative method.The coupled two-level subgrid stabilized Oseen iterative method has two steps.Firstly,on a coarse mesh with mesh size H we use Oseen iterative method of m times to solve subgrid stabilized numerical scheme for the natural convection problem to obtain iterative solution(um H,pm H,Tm H);then by use the iterative solution,solve a linear problem with the Newton itertion of one time on a fine grid with mesh size h ? H to gain the numerical solution(umh,pmh,Tmh).Compared with coupled twolevel subgrid stabilized Oseen iterative method,decoupled two-level subgrid stabilized Oseen iterative method have a few differences.On a fine grid it use Oseen iteration of one time,and in the right hand of numerical scheme Tmhis replaced with iterative solution Tm H.In this way,we just need to solve two linear subproblems,and this two subproblems can be computed in parallel.Thus it can save computation time.Moreover,numerical examples verify the stability and convergence of the new methods.In Chapter 4,for incompressible magnetohydrodynamics(MHD)equations,based on Stokes,Newton and Oseen iterations we structure various two-level iterative methods.For the strong uniqueness condition,we structure three different two-level iterative methods in which on a coarse mesh solving the iterative solution((um H,Bm H),pm H)with mesh size H and finding a correction solution((umh,Bmh),pmh)by solving the Stokes type,Newton type and Oseen type correction problems on a fine grid with mesh size h.Then we theoretically analyze the stability and convergence of two-level iterative methods.Under the weak uniqueness condition,we consider the one-level Oseen iterative method with mesh size h and present the corresponding stability and convergence for the numerical solutions from theoretical analysis and numerical examples.In Chapter 5,for MHD equations,from penalty method and Stokes,Newton and Oseen iteration we structure new two-level penalty methods: The penalty parameter?(0 < ? ? 1)is set as a real number,in the first step,a nonlinear MHD problem is solved on the coarse mesh by using the usual penalty method;In the second step,a linearization MHD problem is solved on the fine mesh by the penalty method based on Stokes,Newton and Oseen iterations with the coarse mesh solutions.Theoretical analysis and numerical examples prove the stability and convergence for two-level penalty methods.