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

Abstract

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order $μ\in (0,1)$ 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~$Ω$, our analysis suggest that the error in $L^2\bigr((0,T),L^2(Ω)\bigr)$-norm is of order $O(k^{2-\fracμ{2}}+h^2)$ (that is, short by order $\fracμ{2}$ 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 $L^\infty\bigr((0,T),L^2(Ω)\bigr)$-norm. Variable time steps are used to compensate the singularity of the continuous solution near $t=0$.