Finite Volume Element Method On A Shishkin Mesh For Solving Convection-diffusion Problem

In this paper we consider the two–dimensional convection-diffusion problemwhere ? is a perturbation parameter.With decreasing gradually of ?,the solution of this problem changes drastically in the boundary layer or inside the boundary layer.In the process of solving this problem,there may occur non-physical oscillation for some traditional numerical methods in rapidly changing zones,so it is difficult to obtain satisfactory numerical results.However,the Shishkin mesh does well in this problem.Shishkin mesh is a piecewise uniform mesh,which is divided into (?) parts of fine mesh at the boundary layer and (?) parts of coarse mesh at the residual area.This subdivision method has made a breakthrough in the numerical simulation of convection diffusion problem.When researching the problem in this paper,we first solve the convection-diffusion equation with the existing bilinear finite element method on a Shishkin mesh,and then we continue to derive format of the biquadratic finite element and the finite volume element method on a Shishkin mesh.Selecting the Shishkin mesh as the original split,let Uhis the trial function space,and the finite element method of convection diffusion problem is to find uh? Uh thatwhereWe construct a dual subdivision corresponding to Th,let Vhis the test function space of dual subdivision,the finite volume method for convection diffusion problem is to find uh? Uh thatwhereBy comparing the numerical results of the bilinear finite element method and the bilinear finite volume element method,it can be seen that the corresponding convergence order of ?-weighted energy norm is close to the first order,so the bilinear finite volume element method is also effective to solve the convection diffusion equation.Afterwards,we study the biquadratic finite element method and the biquadratic finite volume element method on a Shishkin mesh.The numerical results show that the convergence order of ?-weighted energy norm is close to second order when we use the biquadratic finite volume method,and the error estimation of the biquadratic finite volume element method is approximately second order.As we construct the biquadratic finite volume element scheme,we choose two kinds of dual mesh.The subdivision points are quartiles points and Gauss points,the corresponding maximum norm respectively tend to second-order convergence and third-order convergence.In this paper,there are still some problems in the theoretical study about error estimation,and it is temporarily impossible to give a logical proof.