A DIRECT DISCONTINUOUS GALERKIN METHOD FOR TIME FRACTIONAL DIFFUSION EQUATIONS WITH FRACTIONAL DYNAMIC BOUNDARY CONDITIONS^*

This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions. The well-posedness for the weak solutions is studied. A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes, together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh, and then the fully discrete scheme is constructed. Furthermore, the stability and the error estimate for the full scheme are analyzed in detail. Numerical experiments are also given to illustrate the effectiveness of the proposed method.