SOME GRONWALL INEQUALITIES FOR A CLASS OF DISCRETIZATIONS OF TIME FRACTIONAL EQUATIONS ON NONUNIFORM MESHES

We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Gro"\nwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Gro"\nwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Gro"\nwall inequalities are then applied to sub diffusion problems and the time fractional Allen--Cahn equations for illustration.