Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem
A time-fractional reaction-diffusion initial-boundary value problem with a Caputo time derivative of order \(\alpha\in (0,1)\) is considered. New pointwise bounds on certain derivatives of the solution are derived. Prior work on the analysis of the DDG method considered periodic boundary conditions; in the present work, homogeneous Dirchlet boundary conditions are used instead (which in turn renders the analysis more complicated). An \(L1\) finite difference scheme in time and a direct discontinuous Galerkin (DDG) FEM in space are considered. The mesh is uniform in space and graded in time. A discrete \(L2\)-stability result for the considered scheme is derived. The convergence results provide guidance on how to choose an optimal grading of the temporal mesh. Numerical experiments demonstrate that the theoretical convergence results are sharp; the reported numerical errors agree with the theoretical rate of convergence.
time-fractional reaction-diffusion equation
scientific article; zbMATH DE number 6940691