Discontinuous Galerkin methods for fractional diffusion problems

Abstract: 

In this talk, some features of the DG and hybridizable  DG (HDG)  methods will be discussed first. We propose a time-stepping DG method for the numerical solution of fractional sub-diffusion problems. Generic hp-version error estimates will be derived after proving the well-posedness of the approximate solution. By employing geometrically refined time-steps and linearly increasing approximation orders, we show exponential rates of convergence in the number of temporal degrees of freedom. Moreover, for h-version DG approximations on appropriate graded meshes near t=0, we claim that the error is nearly of optimal order the maximum time-step.

For the spatial discretization of our model problem,  we use the HDG method and show optimal algebraic spatial error estimates assuming that the exact solution is sufficiently regular.  Moreover, for quasi-uniform meshes, we obtain a super-convergence result.