Analysis And Numerical Simulation Of Nonstandard Mixed Element Methods

Mixed finite element method plays an important role in the numerical methods for differential equations. In this thesis, we mainly study the splitting positive mixed finite element method and H1-Galerkin mixed finite element method.Danping Yang (in 2001) proposed a new mixed finite element method called the splitting positive definite mixed finite element procedure to treat the pressure equation of parabolic type in a nonlinear parabolic system describing a model for compressible flow displacement in a porous medium. The proposed procedure has the following advantages: the coefficient matrix of the mixed element system is symmetric positive definite; the flux equation is separated from the pressure equation so that we can obtain an approximate solution of the flux function easily. Here, we study the splitting definite positive mixed element method for some evolution equations, and get the following results:●Splitting positive definite mixed finite element methods are discussed for a class of second-order pseudo-hyperbolic equations, depending on the different physical quantities of interest, two kinds of splitting mixed schemes are proposed. The approximate solution ofσ= a(x)Vυorσ= a(x)(▽υt+▽υ), which does not depend on the approximate solution ofυ, can be solved, and the proposed procedures do not need to solve a coupled system of equations. The existence and uniqueness of mixed element solutions for semidiscrete schemes are proved, and error estimates are derived for both semidiscrete and fully discrete schemes.●Introducing two transformations q=υt andσ= a(x)▽υ+b(x)▽υt, and solving ordinary differential equation for▽υ, a new splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. Compared to standard mixed methods, the proposed method has several attractive features:σ, which dose not depend onυand q, can be solved; the coefficient matrix of the equation forσis symmetric positive definite, and the procedure is implemented easily. Error estimates are derived for both semidiscrete and fully discrete schemes, and the existence and uniqueness of mixed element solutions for semidiscrete schemes are proved. Finally, some numerical results are provided to illustrate the effectiveness of our method. It is easy to see that the proposed scheme can be applied to many important evolution equations such as Sobolev equation and pseudo-hyperbolic equation and so on.In 1998, Pani proposed an H1-Galerkin mixed finite element method for parabolic partial differential equation, the proposed one has the following advantages:they are not subject to the LBB consistency condition; the mixed finite element spaces Vh and Wh may be of differing polynomial degrees; moreover, a better order of convergence for the flux in L2-norm is obtained. In this thesis, we apply the H1-Galerkin mixed finite element method to solve some important evolution equations, and propose some new numerical schemes based on H1-Galerkin mixed finite element method, and obtain the following results:●The H1-Galerkin mixed finite element method is studied for three kinds of nonlinear equations (RLW-Burgers equation; Burgers-Huxley equation; SRLW-equation). Error estimates are derived for both semidiscrete and fully discrete schemes for RLW-Burgers equation, and the existence and uniqueness of mixed finite element solutions are proved. Finally, some numerical results are given to illustrate the effectiveness of H1-Galerkin mixed method for three kinds of nonlinear equations.●By introducing a space-time auxiliary variables q= Vut, a new H1-Galerkin mixed finite element method is constructed for the semilinear strongly damped wave equation. We can get the lower equation for space-time direction and the integro-differential mixed systems. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variable are discussed.●So far, the H1-Galerkin mixed finite element method was applied to many second-order evolution equations. However, the H1-Galerkin mixed method for the higher-order evolution equations, especially, for fourth-order evolution equations has not been studied in the literature. In this thesis, we first proposed the H1-Galerkin mixed method for fourth-order evolution equation. For the need of the analysis of theories, we consider the fourth-order parabolic evolution equation. By introducing three auxiliary variables, the first-order system of four equations is formulated, and the H1-Galerkin mixed finite element method for fourth-order parabolic equation is proposed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates are derived for semidiscrete scheme for several space di-mensions, and the stability for fully discrete scheme is proved by the iteration method. Finally, some numerical results are provided to illustrate the effectiveness of our method. Optimal approximate solutions for the scalar unknown, first derivative, second derivative and third derivative are obtained, which can't be derived by the other mixed methods, and the method can be applied to higher-order evolution equations such as the fourth-order hyperbolic partial differential equations.●By introducing two new auxiliary variables, a new numerical scheme based on the H1-Galerkin mixed finite element method for a class of second-order pseudo-hyperbolic equations is constructed. The proposed procedure can be split into three independent differential sub-schemes and does not need to solve a coupled system of equations. Opti-mal error estimates are derived for both semidiscrete and Crank-Nicolson-Galerkin fully discrete schemes for problems in one space dimension. And the proposed method does not require the LBB consistency condition. Finally, some numerical results are provided to illustrate the effectiveness of our method.●An H1-Galerkin expanded mixed finite element method which combines expanded mixed method and H1-Galerkin mixed method is studied for 1-D regularized long wave-Burgers (RLW-Burgers) equation. The formulation not only keeps the advantages of expanded mixed formulation but also keeps the advantages of H1-Galerkin mixed formu-lation. The existence, uniqueness and stability of semidiscrete mixed element solutions are proved, and the optimal error estimates of the scalar unknown, its gradient and its flux for semidiscrete and fully discrete schemes are derived. Finally, some numerical results are given to illustrate the effectiveness of the proposed method.The layout of this thesis is as follows:In chapter I, we introduce the development for mixed finite element methods and the main results of this thesis; Chapter II study two kinds of splitting positive definite mixed element methods for the pseudo-hyperbolic type equation; We propose a new splitting positive definite mixed element method for second-order viscoelasticity wave equation in chapter III; Chapter IV give some numerical results of H1-Galerkin mixed element procedure for three classes of nonlinear evolution equations (RLW-Burgers equation, Burgers-Huxley equation, SRLW equation); By introducing a new auxiliary variable, we present a new H1-Galerkin mixed element method for fourth-order strongly damped wave equation in chapter V; In chapter VI, we first propose the H1-Galerkin mixed element method for the fourth-order evolution equation, and present some numerical results; Chapter VII propose a new splitting H1-Galerkin mixed element method for the pseudo-hyperbolic equation; Finally, we study the H1-Galerkin expanded mixed finite element method for RLW-Burgers equation in chapter VIII.