A hybridizable discontinuous Galerkin method for fractional diffusion problems

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order with . For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time the approximations are taken to be piecewise polynomials of degree on the spatial domain , the approximations to in the -norm and to in the -norm are proven to converge with the rate , where is the maximum diameter of the elements of the mesh. Moreover, for and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for converging with a rate of root log(Th-2/(alpha+1)) h(k+2).