A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients

TL;DR

This paper introduces a discontinuous Galerkin method for numerically solving time fractional diffusion equations with variable coefficients, providing error analysis and demonstrating near-optimal convergence through numerical experiments.

Contribution

It develops a novel piecewise-linear, time-stepping discontinuous Galerkin method combined with standard spatial discretization for fractional diffusion equations with variable coefficients.

Findings

Error in $L^2((0,T),L^2( abla))$-norm is $O(k^{2-rac{}{2}}+h^2)$

Numerical experiments show an optimal $O(k^{2}+h^2)$ error in $L^((0,T),L^2( abla))$-norm

Variable time steps improve accuracy near $t=0$.

Abstract

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order $\mu\in (0,1)$ with variable coefficients. For the spatial discretization, we apply the standard piecewise linear continuous Galerkin method. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval~$(0,T)$ and a spatial domain~$\Omega$, our analysis suggest that the error in $L^2\bigr((0,T),L^2(\Omega)\bigr)$-norm is of order $O(k^{2-\frac{\mu}{2}}+h^2)$ (that is, short by order $\frac{\mu}{2}$ from being optimal in time) where $k$ denotes the maximum time step, and $h$ is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal $O(k^{2}+h^2)$ error bound in the stronger $L^\infty\bigr((0,T),L^2(\Omega)\bigr)$-norm. Variable time steps are used to compensate the singularity of the continuous solution near $t=0$.