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 $k\ge 0$ on the spatial domain~$\Omega$, the approximations to $u$ in the $L_{\infty }(0,T;L_2(\Omega ))$-norm and to $\nabla u$ in the $L_{\infty }(0,T;{\mathbf{L}}_2(\Omega ))$-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 $k\ge 1$ 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 $\sqrt{\log (T{h}^{-2/(\alpha +1)})}{h}^{k+2}$.

References in corpus (1)

• A spectral order method for inverting sectorial Laplace transforms