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 (Formula presented.) with (Formula presented.). For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time (Formula presented.)∈[0,T] the approximations are taken to be piecewise polynomials of degree (Formula presented.) on the spatial domain (Formula presented.), the approximations to u in the (Formula presented.)-norm and to ∇u in the (Formula presented.)-norm are proven to converge with the rate (Formula presented.), where h is the maximum diameter of the elements of the mesh. Moreover, for k≥1 and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate of (Formula presented.).