A hybridizable discontinuous Galerkin method for fractional diffusion problems

TL;DR

This paper develops and analyzes a hybridizable discontinuous Galerkin method for fractional diffusion equations, providing optimal error estimates and superconvergence results for numerical solutions.

Contribution

It introduces a novel HDG method for fractional diffusion problems and derives optimal and superconvergence error estimates under regularity assumptions.

Findings

Optimal algebraic error estimates for the HDG method.

Superconvergence result for k≥1 on quasi-uniform meshes.

Convergence rates of h^{k+1} and h^{k+2} with logarithmic factors.

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 $\sqrt{\log(T h^{-2/(\alpha+1)})}\, \,h^{k+2}$.