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 γ (1 < γ < 2). We establish the stability of this method, and derive the optimal convergence in the H 1 (0, T ; L 2 (Ω))-norm and suboptimal convergence in the discrete L ∞ (0, T ; H 1 0 (Ω))-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 (Ω))norm can still be achieved by using graded grids in the time discretization. Finally, numerical experiments are performed to verify the theoretical results.

Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

“…Proof. As the proof of (8) is analogous to that of (7) and the case γ, β ∈ N is trivial, we only prove (7) for the case that γ / ∈ N or β / ∈ N. We first use the standard scaling argument to prove the case β = 0 and 0 < γ < 1. By definition we have…”

“…In our previous work [6], a time-spectral method for fractional wave problems was designed, which possesses exponential decay in temporal discretization, under the condition that the solution is smooth enough. Recently, to conquer the singularity in time variable, Li et al [7] presented a space-time finite element method for problem (1), and proved that high-order temporal accuracy can still be achieved if appropriate graded temporal grids are adopted. Under some conditions, problem (1) is equivalent to an integro-differential model, and there are many works on the numerical methods for this model; see [9,10,18] and the references therein.…”

Error analysis of interpolated coefficient finite elements for nonlinear fractional parabolic equations

“…They play a very important role in wave propagation, finance, physics, engineering and so on (see, e.g., [1,2] and the references therein). Since the exact solutions of most FPDEs are very difficult to obtain, many numerical methods were presented, such as, finite difference methods [3,4,5], finite element methods (FEMs) [6,7,8,9,10], mixed FEMs [11,12], space-time FEMs [13], finite volume methods [14], spectral methods [15,16,17] and so on.…”

Optimal error estimation of a time-spectral method for fractional diffusion problems with low regularity data

“…Many numerical methods have been developed in the past a dozen years. Among existing works, four types of temporal discretization are most prevailing, i.e., finite difference methods (L-type schemes) [2,24,36,41], convolution quadrature methods [11,13,60,62], finite element methods [27,28,29,31] and spectral methods [26,35,55,64]. Under certain circumstances, problem (1) has an equivalent form like…”