A hybridizable discontinuous Galerkin method for fractional diffusion problems

Abstract

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order $$-\alpha $$-? with $$-1<\alpha <0$$-1<?<0. For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time $$t \in [0,T]$$t?[0,T] the approximations are taken to be piecewise polynomials of degree $$k\ge 0$$k?0 on the spatial domain $$\varOmega $$Ω, the approximations to $$u$$u in the $$L_\infty \bigr (0,T;L_2(\varOmega )\bigr )$$L?(0,T?L2(Ω))-norm and to $$\nabla u$$?u in the $$L_\infty \bigr (0,T;\mathbf{L}_2(\varOmega )\bigr )$$L?(0,T?L2(Ω))-norm are proven to converge with the rate $$h^{k+1}$$hk+1, where $$h$$h is the maximum diameter of the elements of the mesh. Moreover, for $$k\ge 1$$k?1 and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $$u$$u converging with a rate of $$\sqrt{\log (T h^{-2/(\alpha +1)})}\, \,h^{k+2}$$log(Th-2/(?+1))hk+2.