A space-time finite element method for fractional wave problems

In this paper, we analyze a space-time finite element method for fractional wave problems involving the time fractional derivative of order gamma(1 < gamma < 2). We first establish the stability of the proposed method and then derive the optimal convergence rate in H-1(0, T; L-2(Omega))-norm and suboptimal rate in discrete L-infinity (0, T; H-0(1) (Omega))-norm. Furthermore, we discuss the performance of this method when the true solution has singularity at t = 0 and show that optimal convergence rate with respect to H-1(0, T; L-2(Omega))-norm can still be achieved by using graded temporal grids. Finally, numerical experiments are performed to verify the theoretical results.