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

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order μ ∈ (0, 1) with variable coefficients. For the spatial discretization, we apply the standard continuous Galerkin method of total degree ≤ 1 on each spatial mesh elements. 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}left ((0,T),L^{2}({Omega })right ))-norm is (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(k² + h²) error bound in the stronger (L^{infty }left ((0,T),L^{2}({Omega })right ))-norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.