Researches On Finite Element Methods For Several Types Of Nonlinear Fluid Dynamics And Phase-field Models

The analysis of finite element methods?FEMs?for fluid mechanics and phase field models is always a hot issue that attracted lots of concerns.This thesis main-ly focuses innovatively on convergence,superclose and superconvergence analysis of some fully-discrete schemes for several types of nonlinear models which have important physical significance and wide application backgrounds?such as un-steady Brinkman-Forchheimer equation,unsteady Navier-Stokes equation with damping,unsteady natural convection equation,Cahn-Hilliard equation,Allen-Cahn equation,Cahn-Hilliard-Navier-Stokes equation?from different perspectives of conforming FEM,nonconforming FEM,mixed FEM?MFEM?,two-grid FEM and etc,with our own typical analytical skills,such as mean value technique,transfer techniques with respect to the time step?which can be viewed as the discrete derivative transfer?,and taking difference quotient at different time lev-els,etc.Numerical experiments are also designed to verify the theory.The main innovations of the paper are:1)For the unsteady Brinkman-Forchheimer equation,a two-grid algorithm which is much more efficient compared with the traditional MFEM is proposed,and the corresponding optimal error estimates are obtained;for the unsteady damped Navier-Stokes equation,a nonconforming MFE approximation scheme is proposed,the superclose and superconvergence error results for each variable are deduced for the first time;2)The superclose and superconvergence analysis of the linearized backward Euler fully-discrete scheme and the second-order BDF scheme which are not yet covered are discussed for the unsteady natural convection equations.Contrary to the traditional projection-based methods employed in most of the previous literature,the above new techniques are employed to overcome the difficulties brought by the convection term and the couplings between different variables,and then get high-accuracy results.Especially,superconvergence error without the restriction on the ratio between the spatial parameter and the temporal step for the linearized Euler scheme is deduced through the splitting technique;3)Two grid methods?TGMs?based on conforming bilinear element and nonconforming₁FE are proposed for Cahn-Hilliard equation and Allen-Cahn equation respectively,and the superclose and superconvergence results of the variables in the corresponding discrete schemes are obtained.Especially,a more general stability analysis of the numerical scheme for Allen-Cahn equation is derived based on the monotonic character of the nonlinear term in the equation,in which the restriction of8)(6_?|1)^???|?in the previous literature is removed;4)Error analysis of a convex splitting energy stable fully-discrete scheme for the Cahn-Hilliard-Navier-Stokes equation is discussed.The nonconforming FE pair is chosen to approximate the Navier-Stokes equation in the coupled equation system,and the bilinear element to the Cahn-Hilliard equation,the superclose and superconvergence error estimates are derived for each variable for the first time.Firstly,a TGM based on conforming MFEM for the unsteady Brinkman-Forchheimer equation is discussed.Using some conventional estimation tech-niques,optimal error estimates for the variables in the Crank-Nicolson fully-discrete scheme are obtained.Numerical experiments are given to prove that the TGM saves two-third of the calculation amounts compared with the tradition-al MFEM;In addition,the linearized Euler fully-discrete approximation scheme based on the nonconforming FE pair CNR₁/₀is investigated for the unsteady Navier-Stokes equation with damping.The mean value skill and the idea of tak-ing the difference quotient in adjacent time levels are adopted to deal with the difficulties caused by the convection term and the incompressible condition,the superclose results forin the energy norm andin the²norm are derived.Then,the global superconvergence results are also obtained by the interpolation post-processing method.Secondly,the MFEM based on the conforming and nonconforming elements for the unsteady natural convection equation are investigated,which mainly in-cludes three parts.At first,a linearized backward Euler fully-discrete scheme based on the element pair of CNR₁/₀is constructed.Some special tech-niques,such as introducing the mean value skill,the discrete derivative transfer techniques,are employed to handle the convection term?·??and the coupling term·?.Combining with the properties of the chosen elements,the super-close error estimates for each variable are deduced.By using the interpolation postprocessing technique,the global superconvergence properties are obtained subsequently.Then,we still consider the linearized backward Euler scheme in time for the problem.The difference is that the system is discretized by the Bernadi-Rangel element for the velocityand the pressure,and the bilinear element for the temperaturein space.Different from the first part,here the error is split into two parts by introducing a time discrete system.And then the superclose and superconvergence errors without restrictions on the ratio be-tween the spatial parameter and temporal step are analyzed;At last,based on the element pair of₁₁/₀,a linearized second-order BDF fully-discrete scheme is proposed.By introducing some new derivative transfer skills?taking the dif-ference quotient at different time levels and combining the techniques used in the previous two sections,the superclose and superconvergence error results of the variables are also achieved.Then,the TGMs for Cahn-Hilliard equation and Allen-Cahn equation are discussed,respectively.For the fourth-order Cahn-Hilliard equation,the problem is transformed into a second-order coupling problem by introducing another vari-able.The convex splitting fully-discrete scheme is established with the bilinear element,and the corresponding TGM is proposed.Using special techniques,the results of the superclose error results which have not been discussed are obtained for the proposed algorithm.Then,the interpolation postprocessing technique is also used to obtain the global superconvergence results.For the Allen-Cahn equa-tion,the TGM is established based on the nonconforming₁element.The monotonic nature of the nonlinear term helps us to obtain the stability analysis of the numerical scheme without the restriction of8)(6_?|1)^???|?.Combined with the properties of the element,the superclose and global superconvergence results are deduced.Finally,a first-order fully-discrete scheme based on the convex splitting in the Cahn-Hilliard equation is constructed for the Cahn-Hilliard-Navier-Stokes system,where the CNR₁/₀element pair is used to approximate the Navier-Stokes equation and the bilinear element to the Cahn-Hilliard equation in space.To overcome the difficulties brought by the strong nonlinearity and the complex couplings of variables in the system,the combination of these estimate methods in the previous parts is employed to obtain the superclose and superconvergence results for each variable skillfully.It is worthy mentioning that analysis focused on these superconvergence s-tudies,especially for the nonconforming FEMs,are rarely referred in the previous literature.In addition,numerical examples are provided to verify the correctness of the theoretical analysis,the rationality and effectiveness of the corresponding methods in each part.