Discontinuous Galerkin Finite Element Methods For Time-Fractional Partial Differential Equations

In recent years,the study of fractional calculus,especially fractional differential equations,has attracted extensive attention.The singularity and nonlocality of generalizing integer-order calculus to real or even variable orders are well adapted to describe(time)delay or memory and genetic effects.Since explicit analytical solutions of fractional differential equations are usually unavailable,it is very important to design high-precision numerical methods to solve the fractional differential equations.However,when solving the fractional differential equation,we usually encounter two major difficulties,which are also two very important characteristics of the fractional differential operator: 1)the nonlocality of the fractional derivative operator;2)The solution of fractional differential equation is usually singularity at the initial moment.In this paper,we study(local)discontinuous Galerkin methods for Caputo type time-fractional partial differential equations,and construct fully discrete(local)discontinuous Galerkin schemes respectively.The stability and a priori error estimate of the fully discrete scheme are proved strictly.In this paper,we discretize the Caputo derivative by using the L1 scheme on uniform mesh and the L2-1?scheme(as introduced by Alikhanov [4])on non-uniform mesh.Combined with the(local)discontinuous finite element method,we deal with the spatial direction.To solve several kinds of fractional partial differential equations(time-fractional convection equation,time-fractional Burgers equation,time-fractional fourth-order equation,and time-fractional convection-diffusion-reaction equation).The l2-1?method on a nonuniform grid can well deal with the weak regularity of the solution of fractional partial differential equations at the initial moment.The main contents of these equations are as follows:Firstly,we discretize the fractional derivatives of one-dimensional time-fractional convection equations by using the L2-1?method on a nonuniform grid,and combine with the discontinuous Galerkin method in space direction.On this basis,the fully discrete discontinuous Galerkin scheme of the time-fractional convection equation is obtained by constructing appropriate numerical fluxes,the stability and a priori error estimation of the fully discrete numerical scheme are also proved.Finally,two numerical experiments are carried out to verify the previous theoretical analysis and show the validity of the L2-1?method for discretization of time fractional derivatives on non-uniform grids.Secondly,for the one-dimensional time-fractional Burgers equation,we consider the L1 scheme on the uniform grid to discretize the Caputo type time fractional derivative,and then combine with the local discontinuous Galerkin method in the spatial direction.On this basis,the fully discrete local discontinuous Galerkin scheme of the time-fractional Burgers equation is obtained by constructing appropriate numerical fluxes.For the numerical fluxes of the nonlinear terms,we use Lax-Friedriches fluxes to ensure the unconditional stability of the numerical scheme.In addition,we prove a priori error estimate for the linear case.Finally,the correctness of the theoretical analysis is further verified by numerical experiments.Then,we also study the local discontinuous Galerkin method for one-dimensional time-fractional fourth-order equations.The method is also based on the L2-1?method on a nonuniform grid in time discretization and the local discontinuous Galerkin method in space direction.By introducing three auxiliary variables and designing appropriate numerical fluxes,the fully discrete local discontinuous Galerkin scheme for the model problem is obtained.At the same time,we prove the stability and a priori error estimation of the fully discrete numerical scheme.Finally,the theoretical analysis is verified by numerical experiments,and it is shown that the L2-1?method on non-uniform grids is effective for time fractional problems with singularity at the initial time.Finally,for the one-dimensional time-fractional convection-diffusionreaction equation,we also use the L2-1?method on the inhomogeneous grid to discretize the fractional derivative of the equation in order to overcome the weak regularity problem of the equation at the initial moment of solution.The spatial direction is discretized by local discontinuous Galerkin method.By introducing auxiliary variables and constructing appropriate numerical fluxes,a fully discrete local discontinuous Galerkin scheme is obtained.At the same time,we also prove the stability and a priori error estimate of the fully discrete scheme.Finally,several numerical experiments are designed to verify the convergence and effectiveness of the proposed method for Caputo type fractional convection-diffusionreaction equations.