Discontinuous Galerkin Methods For Several Fractional Partial Differential Equations

Fractional partial differential equations have been widely used to describe many abnormal phenomena in nature and engineering technology due to their memory and heredity.Because the nonlocality and complexity structure of fractional differential operators,it is difficult to get the analytical solutions for most fractional partial differential equations.Although we can obtain analytical solutions of only a few equations,these solutions contain some special functions that are hard to calculate generally.Therefore,it is significant to study efficient numerical methods of fractional partial differential equations.In this dissertation,combining with the finite difference approach,several high-order fully discrete discontinuous Galerkin methods are constructed for three kinds of partial differential equations:space fractional partial differential equations,space-time fractional partial differential equations and time fractional partial differential equations.The stability and convergence of these methods are strictly analyzed.The main work includes:The nodal discontinuous Galerkin method for spatial fractional Navier-Stokes equa-tions is constructed.By introducing the auxiliary variables,the original equations are transformed into a first-order linear system.By selecting the numerical fluxes and adding appropriate penalty terms,the nodal discontinuous Galerkin method is designed in the spatial direction,and the spatial semi-discrete scheme is obtained.In temporal direction,a second-order Lagrange method is used to get full scheme.The stability of the fully discrete system is proved,and the error estimation of velocity is given.Then the relation between pressure error and velocity error is established by combining continuous inf-sup condition with discrete space,and the error estimation of pressure is obtained.Two kinds of hybridizable discontinuous Galerkin methods for space-time fractional advection-dispersion equations are proposed.In space,the hybridizable discontinuous Galerkin method is constructed by selecting numerical fluxes carefully.The main idea is reducing the globally coupled unknowns to certain approximations on the element boundaries by a hybridization technique.In time,L1 formula along with the weighted shift Grünwald-Letnikov formula are used to approximate respectively,and two fully discrete numerical schemes are obtained.The stability of two numerical methods are proved and the strict error analyses are carried out.A direct discontinuous Galerkin method for time fractional diffusion equations with fractional dynamic boundary conditions is constructed.By defining special numerical fluxes,a direct discontinuous Galerkin scheme in space is designed.No other auxiliary variables are introduced in the calculation.In order to deal with the initial weak singularity of solution caused by fractional derivative operator,a second-order L2-1scheme is proposed based on graded mesh.The stability is proved,and optimal error estimates are obtained under some conditions.