The Galerkin Space-time Finite Element Methods For Two Kinds Of Fractional Partial Differential Equations

The main contents of this paper are divided into the following two parts:one is to use the time-discontinuous Galerkin space-time finite element method to solve the fractional advection-dispersion equation;the other is to use the variable mesh continuous Galerkin space-time finite element method to solve the space-fractional telegraphy equation.The specific research contents are as follows:In the first part,the traditional Galerkin finite element method and the discontinuous Galerkin finite element method are respectively adopted in space and time for the fractional advection-dispersion equation.The next step is to construct the time-discontinuous space-time finite element format.The approximate solution is expressed as a linear combination of lagrange interpolation polynomials with Radau integration points as nodes,and the space-time format is discretized combined with Radau integral formula.Then the stability of the finite element solution can be proved without adding any restriction conditions to the time and space grids.Further introducing the spatiotemporal elliptic projection operator.By using this operator,we can obtain the proof of the L~∞(L~2)-norm optimal order error estimation for the numerical solution in the fully discrete format.Finally,a numerical example is used to verify the proposed time-discontinuous finite element scheme.In the second part,based on the Galerkin finite element method and the idea of decreasing the order,a variable mesh continuous space-time finite element scheme for the space-fractional telegraphic equation is constructed.Under the condition that any time layer can be divided into different spaces without adding personality constraints to the time grid and space grid.The finite difference and finite element method are used to prove the stability of the spatial and temporal finite element scheme and the L~∞(L~2)-norm optimal error estimation.