A space-time finite element method for fractional wave problems

Abstract

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.

References in corpus (1)

• A time-spectral algorithm for fractional wave problems

Cited by in corpus (1)

• Convergence analysis of a Petrov-Galerkin method for fractional wave problems with nonsmooth data