Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

TL;DR

This paper analyzes the convergence and superconvergence of HDG methods for time fractional diffusion problems, demonstrating optimal and superconvergent rates of approximation for solutions and their gradients.

Contribution

It provides the first convergence and superconvergence analysis of HDG methods applied to time fractional diffusion models with Caputo derivatives.

Findings

HDG approximations converge with rate h^{k+1} for sufficiently regular solutions.

Superconvergence allows a new approximation to converge with rate h^{k+2}.

Numerical experiments confirm the theoretical convergence rates.

Abstract

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order $0<\alpha<1$. For each time $t \in [0,T]$, the HDG approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$\Omega$, the approximations to the exact solution $u$ 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 $h^{k+1}$ provided that $u$ is sufficiently regular, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate $h^{k+2}$ (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.