Researches On Mixed Finite Elements For Some Nonlinear Equations

In this thesis, we consider some kinds of nonlinear equations (including magne-tohydrodynamics (MHD) equations,Stokes equations with damped term,convection-diffusion equations,reaction-diffusion equations and nonlinear parabolic equations, etc), study the nonconforming finite element methods, the least-square nonconform-ing finite element methods, sharp estimates of error constants and two-grids and moving grids for new mixed variational forms, etc, from different points of view and give deep and comprehensive investigations on the construction of the mixed finite element methods,convergence analysis,superclose and supercorivergence, etc.Firstly, a family of low order nonconforming elements (including tetrahedra el-ements and hexahedra elements) are used to approximate a nonlinear, fully coupled stationary incompressible MHD equation in 3D. When the magnetic field belongs to H1(Ω)3 and H(curl;Ω), the nonconforming mixed finite element methods are ana-lyzed for the MHD equations discussed in [14] and [27], respectively. The existence and uniqueness of the approximate solutions are proved and the optimal error es-timates about the corresponding unknown variables are given. Furthermore, a new approach is adopted to prove the discrete Poincare-Friedrichs inequality.Secondly, we apply the constrained nonconforming rotated Q1 element and the piecewise constant element to approximate the velocity and the pressure for the stationary.. impressible Stokes equations with damped term, respectively. The existence and uniqueness of the approximated solutions are proved. Employing the prior estimates of the exact and approximate solutions and choosing the appropriate parameters a, v and r, the optimal error estimates and the superclose results are derived. Finally, the O(h2) global superconvergence in H1-norm for the velocity and L2-norm for the pressure is obtained by use of a postprocessing technique. Thirdly, we analyze the least-square nonconforming finite element scheme and its two modified forms for the convection-diffusion equations. The rectangular EQ1rot element and zero order R-T element are used to approximate to the displacement and the stress. The existence and uniqueness of the approximate solutions are proved by means of some special properties of the elements. The O(h) order error estimates for the stress in H(div)-norm and the displacement in broken H1-norm are derived. At the same times, we use zero order R-T element and P1 triangular element to discrete the equations by the least-square finite element methods, we get sharp estimations of error constants in H(div)-norm for the stress and L2-norm for the displacement under right angled triangular meshes. Furthermore, the corresponding numerical experiments are carried out to verify the theoretical analysis.Then, we construct a new mixed variational form for the semilinear reaction-diffusion equations by lowest rectangular conforming mixed finite elements. The error convergence order of O(Δt+h2) in L2-norm about the corresponding unknown functions are derived in the fully-discrete schemes by use of the special properties of elliptic projections and one order is improved than that of [136] using the traditional mixed variational form. Based on two grids algorithms, we derive the meshes ratio H= O(h1/3) and the convergence order O(Δt+h+H3) under the condition of two steps iteration, in order to obtain the results, the literature [136] need iterative three steps. The meshes ratio H= O(h2/9) and the convergence order O(Δt+h+H9/2) are obtained in this chapter under the condition of three steps iteration. Thus, we achieve the purpose of iterating the same step number, increasing the meshes ratio and reducing computation costs.Finally, we use the nonconforming EQ1rot clement and zero order R-T element and construct a new mixed scheme for nonlinear parabolic equations with moving grids. By use of the special property of the element, i.e., its consistency error is one order higher than the interpolation error in the broken energy norm, the convergence analysis is presented and the optimal order error estimates are derived. The results of this chapter can be extended to any convergence order by conforming mixed finite element approximations.