The Study Of Nonstandard Mixed Finite Element Methods For Convection-diffusion Equations

Convection-diffusion equation arises in many scientific and engineering applications,such as energy development,electronic science,pollution of chemical solute and flow movement of porous media.It can be encountered frequently in real life,so the solution of its mathematical models has become an important issue.Compared to the traditional mixed finite element methods,on one hand,both the stabilized mixed finite element method andH~1-Galerkin mixed finite method can essentially solve the difficulties in choosing the mixed finite element spaces.On the other hand,its coefficient matrix is symmetric and positive definite.The method of lumped masses as a modified finite element method has less computational costs with no effect on convergence precision.Taking the above advantages,in this paper,we focus on the following three aspects:Firstly,for the convection-diffusion-reaction equation,we combined the stabilized mixed finite element method and the method of lumped masses in the space region.And in the time region we adopted the backward Euler discretization.This forms a new fully discrete scheme which is called Euler-type lumped mass/stabilized mixed finite element scheme.Convergence analysis is then discussed,and finally numerical experiments are given.Secondly,we still considered the stabilized mixed finite element method and the method of lumped masses in the space region.But in the time region we discussed Crank-Nicolson method.And we got a new fully discrete scheme which is called C-N-type lumped mass/stabilized mixed finite element scheme.Then we gave its error analysis and numerical experiment.Finally,we discussed the combination ofH~1-Galerkin mixed method and the method of lumped masses.And we used the Crank-Nicolson method to discrete the time domain.This forms a new scheme which is calledH~1-Galerkin-type lumped mass finite element scheme.Also,we show the corresponding convergence analysis of its discrete schemes.Numerical experiment is given to verify the effectiveness and feasibility of the new schemes.