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 \(-1<\alpha<0\). For exact time-marching, we derive optimal algebraic error estimates {assuming} that the exact solution is sufficiently regular. Thus, if for each time \(t \in [0,T]\) the approximations are taken to be piecewise polynomials of degree \(k\ge0\) on the spatial domain~\(\Omega\), the approximations to \(u\) in the \(L_\infty\bigr(0,T;L_2(\Omega)\bigr)\)-norm and to \(\nabla u\) in the \(L_\infty\bigr(0,T;{\bf L}_2(\Omega)\bigr)\)-norm are proven to converge with the rate \(h^{k+1}\), where \(h\) is the maximum diameter of the elements of the mesh. Moreover, for \(k\ge1\) 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 \(\sqrt{\log(T h^{-2/(\alpha+1)})}\, \,h^{k+2}\).