Research On Local Discontinuous Finite Element Method For Convection-Diffusion Equations

As a further extension of the finite volume method and the finite element method,discontinuous Galerkin methods are a class of efficient numerical solutions for solving hyperbolic equations.For high order partial differential equations,firstly,the auxiliary variables are established to convert them into first order equations,and then the equations are calculated by using the discontinuous Galerkin method,which is called the local discontinuous Galerkin method.Both of them have high order convergence accuracy and are efficient and stable modern numerical algorithms.By selecting the order of the polynomial used to approximate the exact solution in the discontinuous finite element space,we can make the convergence error of the numerical solution to the exact solution have an arbitrarily high order.In addition,for the case where the solution has a discontinuity,the numerical solution can be better controlled near the discontinuity point by selecting an appropriate slope limiter,not only the oscillation can be eliminated,but also a steep state can be made near the break.Our attention is mainly paid to the stability and error estimation of the local discontinuous Galerkin method for one-dimensional variable coefficient linear convection-diffusion equations with periodic boundary conditions.The stability ensures that the numerical solution obtained from the scheme can be controlled by its initial value and will not deviate seriously with the development of time.The order of convergence is one of the important criteria to measure whether a numerical algorithm is excellent.A good algorithm should have high order accurate and high resolution,which is the embodiment of the similarity between numerical solution and exact solution.In this paper,the stability and convergence conclusions of the local discontinuous Galerkin method for one-dimensional variable coefficient linear convection-diffusion equations are given,which further strengthens the theoretical basis of the local discontinuous Galerkin method for solving the variable-coefficient partial differential equations.We proceed to propose and analyze the stability of the convection-diffusion equation which has a convection term with a variable coefficient function and a diffusion term with a constant coefficient and another equation which has a convection term with a constant coefficient and a diffusion term with a variable coefficient function.By combining the results of the above two parts,the stability conclusions of the one-dimensional variable coefficient linear convection-diffusion equation under periodic boundary conditions are given.The key in the stability proof process is the selection of numerical fluxes.In this paper,the general numerical flux is selected for the establishment of the LDG scheme.The matching of the convection section flux parameter with the value of the variable coefficient function at the boundary point of each splitting unit is the key to maintaining a stable scheme.By constructing a special slice global projection,and giving the proof of its existence uniqueness and the optimal approximation property in a form of theorem,we end by studying the error estimate of local discontinuous Galerkin methods for one-dimensional variable coefficient linear convection-diffusion equations on Cartesian meshes.Here and below,k?1 is the piecewise polynomial degree of the finite element space.Combining with the Gauss-Radau projection and the standard local ~2L projection,the error estimate of the convection-diffusion equation which has a convection term with a variable coefficient function and a diffusion term with a constant coefficient and another equation which has a convection term with a constant coefficient and a diffusion term with a variable coefficient function were carried out,respectively.The analysis results showed that the convection-diffusion equation has LDG scheme up to k order convergence,while the convection term takes a variable coefficient function and the diffusion term takes a constant coefficient,and the convection-diffusion function has LDG scheme up to k(10)1order convergence,while the convection term takes a constant coefficient and the diffusion term takes a variable coefficient function.Combining the proof process of the above two parts,this paper gives the conclusion that the numerical solution obtained by the local discontinuous Galerkin method of one-dimensional variable coefficient linear convection-diffusion equation with periodic boundary conditions is k order convergence when general numerical fluxes are used.Last but not the least,based on the theory of stability analysis and convergence analysis,three representative numerical examples are selected for numerical experiments.According to the calculation results,the local discontinuous Galerkin method in choosing general numerical fluxes for solving variable coefficient linear convection-diffusion equations can obtain a degree k(10)1 of convergence,this is the optimal convergence order for solving high order equations by choosing this kind of numerical methods.In summary,the local discontinuous Galerkin method is a numerical algorithm with high order accurate and high resolution for solving high order partial differential equations.In this paper,we studied the stability and convergence of a one-dimensional variable coefficient linear convection-diffusion equation with this method when general numerical fluxes are considered.The theoretical basis further illustrates the high precision characteristics of this method in solving high order equations.At the same time,the numerical examples are used to provide the factual basis for the theoretical results.It is powerful to demonstrate that the local discontinuous Galerkin method has computational stability and superiority for the variable coefficient equation.