High Efficient Finite Element Methods For The Incompressible Coupled Navier-Stokes Equations

Today,scientific computation which is as important as theory and experiment has become the third scientific research method with the rapid development of computer science.In many cases,computational simulation will become the only or primary method.In this thesis,we will study coupled systems of the incompressible Navier–Stokes equations and temperature equations(or Maxwell's equations).Directly solving and computing will encounter some difficulties since these systems are complex nonlinear coupling problems and people's understanding to its essence is still very limited at present.Importantly,it is very difficult to find their exact solutions.Therefore,it is very urgent to construct and study the efficient numerical algorithms to solve these problems.In this thesis,we will consider the natural convection(NC)equations and Magnetohydrodynamics(MHD)equations and construct several stable and high efficient finite element methods combining with the difficulties in the process of solving equations.Firstly,based on the basic idea of two level method,we construct a novel efficient two step algorithm for solving the NC problem.The basic idea of the algorithm is to compute an initial approximation for the velocity,pressure and temperature based on a lowest equal-order finite element pair,i.e.,(P1,P1,P1),then to solve a linear system based on a quadratic equal-order finite element pair,i.e,(P2,P2,P2)on the same mesh.The two step method can avoid the discussion on relation of the coarse and fine meshes,because the method only needs one mesh size,which is different from the two-level or two-grid method.It is noted that the finite element pairs in the part we choose are not satisfied with the discrete inf-sup condition,so stabilized methods based on the local Gauss integration projection method are considered.Moreover,the theoretical analysis and numerical examples show that the two step algorithm not only can achieve the numerical accuracy but also can save a large amount of CPU time compared with the quadratic equal-order stabilized method.Secondly,the coupled Navier–Stokes problems are nonlinear systems,so it need to deal with nonlinear terms in the process of calculation.If we use iterative methods to deal with them,the computation will be doubled per iteration.Thus,we propose an efficient characteristic variational multiscale(C-VMS)method to solve the nonstationary NC problems.The C-VMS method does not need nonlinear iteration with the same accuracy,so it can save a lot of CPU time.Importantly,the C-VMS method can be used to solve the high Rayleigh number problem.Some numerical examples show that the C-VMS method is efficient,reliable and can save a lot of CPU time for this problem.Thirdly,as we know that the main difficulty is the coupling of the velocity and pressure through the incompressibility constraint in the process of solving Navier–Stokes equations.To overcome this difficulty,many scholars have proposed and developed the projection methods.More importantly and appealing,using projection methods,one only needs to solve a sequence of decoupled elliptic equations for the velocity and the pressure at each time step,making it very efficient for large scale numerical simulations.As projection methods do not need iterations,it can save a great deal of CPU time.Based on these,pressure-correction projection finite element methods are proposed to solve nonstationary NC problems in this thesis.The first-order and second-order backward difference formulas are applied for time derivative,the stability analysis and error estimates of the semi-discrete schemes are presented using energy method.These schemes are unconditionally energy stable,so we can use them to solve the high Rayleigh number problem.Ample numerical results are presented to demonstrate the effectiveness of the pressure-correction projection methods for solving these problems.Then,we consider NC problems with large temperature difference.The above three parts of our research work for NC problems are based on the Boussinesq approximation.So far,most of the studies are based on the Boussinesq approximation,however,in most geophysical flows,the NC is often driven by large temperature differences leading to considerable density variations and in these cases the Boussinesq approximation fails,however,papers researching the NC problem with variable density are rare.Therefore,for the study of NC problem with large temperature difference is increasingly urgent.Based on the above three parts of our research work,we find that projection methods are unconditionally stable and highly efficient methods for solving incompressible coupled Navier–Stokes equations.This thesis constructs two novel unconditional stable Guage–Uzawa schemes for solving NC problems with variable density.Furthermore,stability analyses and ample numerical tests are presented to demonstrate these schemes are suitable and high efficient for handling NC problems with variable density.Finally,based on the the previous work,we developed the novel efficient two step algorithm to solve stationary incompressible MHD equations,while,it is slightly different from the first part.Using a lower order finite element pair(P1b,P1,P1)to compute an initial approximation,that is using the Mini-element(P1b,P1)to approximate the velocity and pressure and P2 element to approximate the magnetic field,then applying a higher order finite element pair(P2,P1,P2)to solve a linear system on the same mesh.Note that,the Mini-element and Taylor-Hood element satisfy the discrete inf-sup condition,so it doesn't need stabilized strategy.Lastly,theoretical analysis and numerical experiments show that our method is efficient and reliable.