The Study Of Stabilized Mixed Finite Element Methods

Partial differential equations play an important role both in theory and practical application,so many mathematicians,physicists and engineers pay more and more attention to the numerical solution.Darcy flow equation is also a very practical equation,it can be used in many ways,such as in civil engineering,geological engineering,petroleum exploration and seepage mechanics,etc..Until now,the Darcy flow equation is still valuable and significative.As we all know,the traditional mixed finite element method and finite element method still have some disadvantages.First,the finite element fuction spaces for the state and flux state have to be specially chosen such that they strictly satisfy the LBB consistency condition;Second,the triditional method is computationally cost.In this paper,we will propose a stabilized mixed finite element method and use this method to deal with some practical problems.This paper consists of three parts.First,we will use the stabilized mixed finite element method to deal with the reaction-diffusion equation.We wil provide the corresponding theoretical proof and give a priori error estimates for the stabilized mixed finite element scheme.In addition,we will provide a number of numerical experiments to prove the feasibility and efficiency of the stabilized finite element method and to support the accuracy of our prior error estimates.Second,we will propose two stabilized mixed finite element scheme to solve the unsteady reaction-diffusion equation,where Euler method and the Crank-Nicolson method are used to deal with to the time derivative part,and stabilized method is considered for the discretization of the space denivative part.Then we will provide a priori error estimate for these two schemes.At the end,we will give the corresponding numerical tests to determine the merits of the two methods.Third,we will use the stabilized mixed finite element method to deal with Darcy flow equation,and construct the corresponding numerical schemes.Finally we will also providethe corresponding theoretical proof and a priori error estimates.