Mixed Finite Volume Element Methods For Three Kinds Of Evolution Equations

Mixed finite volume element methods were first introduced in 1995 by Russell for solving a class of second order elliptic problem.Due to the combination of finite volume element method and mixed finite element method,this method can use the advantage of finite volume element method to calculate multiple physical quantities simultaneously,and has been developing rapidly since it was introduced.In this thesis,we apply mixed finite volume element methods to solve two kinds of nonlinear evolution equations and one kind of time fractional reaction-diffusion equation.In Chapter 1,the research status of mixed finite volume element method and three kinds of evolution equations are briefly introduced.In Chapter 2,the semi-discrete,nonlinear backward Euler and linear backward Euler fully-discrete mixed finite volume element schemes for RLW-Burgers equation are constructed by introducing flux function and transfer operator,and the existence and uniqueness of discrete schemes solutions are proved by applying the properties of transfer operator,then,the optimal order error estimates for semi-discrete scheme and fully-discrete scheme are given.In Chapter 3,the semi-discrete mixed finite volume element scheme for Burgers equation is constructed,and the existence,uniqueness,stability and optimal order error estimates of semi-discrete scheme solutions are derived.Then,the nonlinear backward Euler fully-discrete mixed finite volume element scheme for Burgers equation is constructed,the existence and uniqueness of fully-discrete scheme are given by using Brouwer' fixed point theorem,and the stability and optimal order error estimates of fully-discrete scheme solutions are derived.In Chapter 4,the mixed finite volume element scheme for one dimensional time fractional diffusion reaction equation is formulated,stability of discrete scheme is derived,and error estimates are given by introduced generalized mixed finite volume element projection.Numerical examples of three kinds of model equations are given to illustrate the feasibility and effectiveness.