Discontinuous Galerkin Finite Element Methods For Fractional Partial Differential Equations

Fractional calculus(including fractional integral and fractional derivative)is a gener-alization of classical integral calculus,which has almost the same history.In recent years,the singularity and nonlocality of fractional calculus operators has been found to be very suitable for describing materials and processes with memory or genetic properties and long-term interactions.As a consequence,more and more scholars have paid attention to this field.In this paper,several kinds of time-fractional partial differential equations are stud-ied via the finite difference methods/the discontinuous Galerkin(DG)finite element meth-ods,including time-fractional convection equation,time-fractional convection-diffusion-reaction equation,time-fractional Stokes equations,and time-fractional Oseen equations.The main contents are introduced in the following five chapters(Chapters 2-6).In Chapter 2,we study the DG method for the one-dimensional time-fractional con-vection equation,where the time fractional derivative is in the sense of Caputo.Since the solution of the time-fractional differential equation often has weak regularity at the initial time,we use the L1 method on non-uniform meshes to discretize the time direction and use the DG method in the spatial direction.The derived numerical scheme is stable and conver-gent in the sense of L2-norm.Numerical examples illustrate the efficiency and theoretical accuracy of the scheme.Chapter 3 is devoted to studying the DG method for the two-dimensional Caputo-type time-fractional convection equation.Similar to the idea in Chapter 2,we discretize the time-fractional derivative by the L1 method on non-uniform meshes and the spatial derivative by the DG finite element methods including rectangular elements and triangular elements.The stability and convergence of the scheme are both carefully investigated.Finally,several numerical experiments are given to verify the theoretical analysis.In Chapter 4,the exi,stence,uniqueness,and regularity of the solution of one-dimensional Caputo-type convection-diffusion-reaction equation are directly studied.We apply the L1 method on uniform and non-uniform meshes to approximating the temporal fractional derivative,and apply the local discontinuous Galerkin(LDG)method on uni-form meshes to approximating the space derivative.Then we prove that the schemes are stable and convergent in L2-norm.Numerical results are also displayed which support the theoretical analysis.In Chapter 5,we consider the numerical methods for the two-dimensional Caputo-type time-fractional Stokes equations.If the solution u(x,t)(velocity)of the equations is sufficiently smooth with respect to t ?[0,T],for example,u(x,·)?C2[0,T],we use the L1 method on uniform meshes to approximate the time fractional derivative and the LDG method to approach the spatial derivative.With suitable numerical fluxes,we prove that the scheme is stable and obtain the L2 optimal error estimates for the velocity,the stress(gradient of velocity)and the pressure.If the solution u(x,t)of the equations has a certain weak singularity at the initial time,we use the L1 method on non-uniform meshes to discretize the time direction and the LDG method as spatial discretization.We also show that the corresponding scheme is stable.Based on the stability analysis,we obtain the L2 optimal error estimate for the velocity.Finally,several numerical experiments are given to validate the theoretical results.In Chapter 6,we propose a LDG method for the two-dimensional Caputo-type time-fractional Oseen equations.According to the regularity of the solution of the equation at the starting time,we use the L1 methods on uniform and non-uniform meshes to approximate the time fractional derivative and the LDG method to approximate the spatial derivative.The stability analysis and error estimates for both situations are carefully investigated.Nu-merical examples further show that the theoretical analysis is correct.The last chapter concludes the main research contents and presents the possible topic in the future.