A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition

L1 scheme on graded mesh for subdiffusion equation with nonlocal diffusion term

“…Authors in [15] have pointed that due to initial singularity of the solution and the discrete convolution form in numerical Caputo derivative, the traditional H 1 (Ω)-norm analysis (corresponding to the case for a classical diffusion equation) to the time fractional diffusion problem always leads to suboptimal estimates. A similar conclusion is also drawn in [25], where authors have used direct discontinuous Galerkin method for solving the time fractional diffusion equation. For the derivation of optimal error estimate in H 1 (Ω) norm, we follow the idea given in [15,6].…”

“…In the following theorem we show that solution of ( 21) is equivalent to the solution of ( 24)- (25).…”

“…Proof. Suppose (U n , d) is a solution of ( 24)-( 25), then d = l(U n ) and putting this in (25), we get…”

Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction–Diffusion Equation with Robin Boundary Conditions

“…One of the novelties of this paper is mitigation of the difficulty caused by Robin boundary conditions in the discretisation process. At present, there are many papers consider global convergence of the time fractional problem with Robin boundary conditions [10,21,22]; But no local in time error analysis for multi-term timefractional problems with Robin boundary conditions has been considered. This is our motivation for completing this paper.…”