Researches On High-Accuracy Mixed Finite Element Methods

In this dissertation, we focus on the pseudo-hyperbolic equations, strongly damped wave equations, fourth-order parabolic type equations, the convection-dominated diffu-sion equations and the Stokes equations, study nonconforming splitting positive definite mixed finite element method (MFEM), H1-Galerkin MFEM, a new characteristic MFE scheme, stabilized MFEM and modified penalty combinational algorithm from different points of view, and give comprehensive and in-depth studies of supercloseness properties, the global superconvergence and extrapolation of these algorithms which are few involved in previous literature.Firstly, splitting positive definite MFE semi-discrete and fully discrete schemes are investigated for pseudo-hyperbolic equations by choosing nonconforming rectangular quasi-Wilson element. By using of the special property of the element, i.e., its consistency error is two order higher than the interpolation error, and the technique of interpolation opera-tors instead of projections, Superconvergence results in‖·‖div,h norm for the flux variable and optimal order error estimates in L2norm for original variable are derived under almost uniform meshes.Secondly, we mainly consider the high-accuracy analysis of the H1-Galerkin MFEM. On the one hand, some new asymptotic error expansions and extrapolations of H1-Galerkin MFEM for strongly damped wave equations are studied. The main terms of the error be-tween the exact solution and its FE interpolation are determined through a special com-bination of the two adjacent elements and Bramble-Hilbert lemma. In the process, the restriction of zero boundary condition of the approximation space of real stress variable can be removed which is the sufficient condition for getting high-accuracy asymptotic expansions in [90]. With the help of interpolation postprocessing technique and Richard-son extrapolation schemes, the convergence rates with order O(h3), which are two order higher than the error estimates in [96,97] etc. are obtained for both the original variable in H1norm and the actual stress variable in H(div;Ω) norm. A numerical experiment is presented to show the effectiveness of the algorithm. On the other hand, a nonconform-ing H1-Galerkin MFE scheme for strongly damped wave equations is discussed. The expanded nonconforming rotated Q1(EQ1rot) and quasi-Wilson elements are chosen to approximate the original and the real stress variables, respectively. By virtue of interpo-lation operators instead of the projections of the two variables, the supercloseness and superconvergence results for the two variables are derived under almost uniform meshes, which are one order higher than the error estimates in the previous literature [96,97] etc. At the same time, some numerical results are given to confirm the theoretical analy-sis. Then, by introducing three auxiliary variables, the fourth-order parabolic differential equations are reduced to a first-order system of four equations. An H1-Galerkin MFE scheme is constructed to solve the first-order system based on the conforming linear tri-angular element. The supercloseness properties and superconvergence for the above four variables are got, which are one order higher than the error estimates in [100] under the same regularity of solutions.Thirdly, a characteristic nonconforming MFEM is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation, in which the noncon-forming EQ1rot element is chosen as approximation space for original variable, the low-est order Raviart-Thomas element for auxiliary variable, respectively. By employing the high-accuracy properties of the elements, the optimal order error estimates for the two variables are obtained. While, the the error estimates derived from the previous traditional MFE scheme were suboptimal [4,30,32,164]. Furthermore, we give some numerical re-sults to show the effectiveness the new scheme.Then, based on Clement interpolation constructed in stabilized term, a new stabilized MFE scheme of Stokes equations by using of the constrained quadrilateral nonconform-ing rotated Q1(CNQ1rot)-Q0finite element pair is constructed. This proposed scheme has the same attractive computational properties as the stabilized MFEM with local polyno-mial pressure projection in [7,57], moreover the one order higher approximation accuracy of Clement interpolation than that of local polynomial pressure projection guarantees the supercloseness and superconvergence analysis. The existence and uniqueness of the ap-proximate solutions are proved. By virtue of the special characters of the elements and interpolation postprocessing, the superconvergence results with order O(h2) are obtained in broken (H1)2norm for the velocity, and in L2norm for the pressure.Lastly, by a linear combination of two solutions gained by classical penalty FEM for Stokes equations, a modified penalty combinational scheme of low order nonconforming MFE is studied. The O(h2+λmλn) order error estimates are obtained in broken (H1)2norm for the velocity, and in L2norm for the pressure. Compared with the classical penalty one, our method can achieve the high convergence rates with a large penalty parameter, which effectively avoids instability problem for the penalty method resulting from the use of a small parameter.