A space-time finite element method for fractional wave problems

This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $\gamma$ ($1<\gamma<2$). We establish the stability of this method, and derive the optimal convergence in the $H^1(0,T;L^2(\Omega))$-norm and suboptimal convergence in the discrete $L^\infty(0,T;H_0^1(\Omega))$-norm. Furthermore, we discuss the performance of this method in the case that the solution has singularity at $t= 0$, and show that optimal convergence rate with respect to the $H^1(0,T;L^2(\Omega))$-norm can still be achieved by using graded grids in the time discretization. Finally, numerical experiments are performed to verify the theoretical results.