Mixed Finite Volume Element Methods And Numerical Simulation For Evolution Type Equations

Mixed finite volume element methods which combine mixed finite element methods with finite volume element methods, also called as mixed covolume methods, were first introduced by Russell in1995for solving a class of second-order linear elliptic type prob-lem. Soon afterwards, Cai and Jones et al illustrated the effectiveness of the proposed methods, Chou and Kwak et al had done a lot of work in the aspect of theoretical analysis. The methods have the following advantages:the methods use the idea of mixed methods and introduce auxiliary variables (such as gradient function, flux function etc.) to refor-mulate higher order problems as lower order problems, and then reduce the regularity requirement of the finite element spaces; it is easy to handle the complex regions and boundary conditions, and the methods have the simple format similar to finite volume element methods; the definition of the methods uses mixed variational form which is help-ful to make theoretical analysis; the computational effort is less than mixed finite element methods, and convergence accuracy is the same as the mixed finite element methods; the local conservation properties of some physical quantities (such as Mass, momentum etc.) are maintained. Based on the above features and advantages, mixed finite volume element methods have become the important numerical methods for solving differential equations.Rui and Lu (in2005) combined expanded mixed methods with finite volume element methods, and constructed an expanded mixed finite volume element method based on the rectangular grids for a class of second-order elliptic problem. This method Inherits the advantages of expanded mixed methods and finite volume element methods, and can simultaneously calculate three unknown variables. The study of this method is mainly based on rectangular grids, however, the analysis on triangular grids is relatively few. In this thesis, we apply expanded mixed finite volume element method based on triangular girds to solving a class of Sobolev equation with convection term, and give the error analysis and numerical simulation.In recent years, with the need to solve complex practical problems of mathematical physics, numerical methods have higher requirements on the computational efficiency. In this thesis, we simplify the expanded mixed finite volume element methods by combining with splitting idea, and propose a new splitting expanded mixed finite volume element. In the numerical simulation process, this method can first solve the coupling system of two equations to get the numerical solutions of the two variables, and then solve the third equation to get the numerical solution of the third variable, thus, greatly reduce the size of linear equations and the computational time.In this thesis, we will apply the mixed finite volume element methods, expanded mixed finite volume element methods and splitting expanded mixed finite volume element methods to studying some evolution type equations from the aspects of the theoretical analysis and numerical calculation. Due to the different characteristics of each class of equations, the numerical formats we constructed are not the same, and the corresponding theoretical analysis and numerical experiments need to be carried out according to the characteristics of each class of equations. In Chapter1, the features and development status of the mixed finite element methods and mixed finite volume element methods are introduced briefly.In Chapter2to Chapter4, we apply the mixed finite volume element method to solv-ing three classes of evolution type equations. In Chapter2, a mixed finite volume element method for one-dimensional regularized long wave equation is studied. The semidiscrete, nonlinear and linear backward Euler fully discrete schemes for the proposed problem are constructed by introducing the transfer operator based on one-dimensional grids. The optimal error estimates for three schemes are derived by using elliptic projection and L2-orthogonal projection operators. Finally, a numerical example is provided to verify the ef-fectiveness and convergence rate. In Chapter3and Chapter4, we apply the method based on triangular grids to numerically solving a class of two-dimensional pseudo-hyperbolic type equation and nonlinear damped Sine-Gordon equation. By using the lowest order Raviart-Thomas space and piecewise constant function space as the solution function spaces, and introducing the transfer operator γh which maps the lowest order Raviart-Thomas space into the test function space, both the semidiscrete and time-in-implicit fully discrete mixed finite volume element schemes are formulated for two proposed equations, respectively. Optimal error estimates are derived by introducing generalized mixed finite volume projection. Finally, some numerical results for two given equations are provided to test the effectiveness of the proposed schemes. In Chapter5, an expanded mixed finite volume element procedure for a class of Sobolev equation with convection term is formulated by combining the expanded mixed methods with finite volume element methods. This method introduces two auxiliary vari-ables λ=-(>)u and σ=-(a(?)u+b(>)ut) to rewrite the original equation as the system of first-order differential equations, uses the lowest order Raviart-Thomas space as the solution function spaces of the variables λ and σ, and uses the piecewise constant func-tion space as the solution function space of the variable u. Then, both the semidiscrete and backward Euler fully discrete expanded mixed finite volume element schemes are con-structed based on triangular grids by using the transfer operator γh. The existence and uniqueness of semidiscrete scheme solutions are proved by applying the theory of differ-ential equations, and the optimal error estimates for the semidiscrete and fully discrete schemes are derived by using the properties of transfer operator and expanded mixed finite volume projection. Finally, a numerical example is given to illustrate the feasibility and the correctness of the theoretical results.In Chapter6, we introduce two auxiliary variables λ and σ defined in Chapter5, and construct a new splitting expanded mixed finite volume element scheme for the Sobolev equation with convection term. The difference between this scheme and the expanded mixed finite volume element scheme is in that:expanded mixed finite volume element scheme needs to solve the coupling system of three equations, and then the size of the coefficient matrix in the process of numerical calculation is relatively large; however, the expanded splitting mixed finite volume element scheme can first solve the coupling system of two equations to get the numerical solutions of the variables λ and σ, and then solve the third equation to get the numerical solution of the variable u, thus, greatly reduce the size of linear equations and the computational time. Finally, some numerical results are given to illustrate the feasibility and the correctness of the theoretical results.In Chapter7, a splitting expanded mixed finite volume element method for a class of parabolic type integro-differential equation is formulated. By using transfer operator γh and introducing auxiliary variables λ(x,t)=-(?)u(x,t) and σ(x, t)=-(a(x)(?)u(x,t)+∫0tk(x, t,τ)(?)u(x, τ)dτ), both the semidiscrete and backward Euler fully discrete splitting expanded mixed finite volume element schemes are constructed. In the fully discrete scheme, left rectangle quadrature rule and backward Euler scheme are applied to getting the discretization of the integral term and the time derivative term, respectively. The optimal error estimates for two schemes are derived by introducing the Volterra type expanded mixed finite volume projection and using the properties of the transfer operator. Finally, a numerical example is given to confirm the theoretical results.