Time-stepping discontinuous Galerkin methods for fractional diffusion problems

Time-stepping $$hp$$hp-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order $$-\alpha $$-α with $$-1<\alpha <0$$-1<α<0 will be proposed and analyzed. Generic $$hp$$hp-version error estimates are derived after proving the stability of the approximate solution. For $$h$$h-version DG approximations on appropriate graded meshes near $$t=0$$t=0, we prove that the error is of order $$O(k^{\max \{2,p\}+\frac{\alpha }{2}})$$O(^{kmax{2,p}+α2}), where $$k$$k is the maximum time-step size and $$p\ge 1$$p≥1 is the uniform degree of the DG solution. For $$hp$$hp-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.