2015/10/31 by Mustapha, K., Abdallah, B., Furati, K. M. +1
#FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1511.00163
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 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 L2((0,T),L2(Ω))-norm is of order O(k^2-\fracμ2+h2) (that is, short by order \fracμ2 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^∞((0,T),L2(Ω))-norm. Variable time steps are used to compensate the singularity of the continuous solution near t=0.