Time two-mesh algorithm combined with finite element method for time fractional water wave model

Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

“…To improve the computation efficiency of the FE discrete system (3.15) and (3.16), we consider the following TT-M system based on FE method, which includes the time coarse mesh τ c and the time fine mesh τ , see Ref. [9] Step I: Firstly, we get the coarse time mesh numerical approximation U n C by equations (3.15) and (3.16), i.e.…”

“…Remark 3.5 (i). Fast TT-M algorithm is proposed by solving the time fractional PDE in [9], and here for the first time applied to the nonlinear space fractional PDE.…”

“…Jiang et al [36] proposed a fast method of the time Caputo fractional derivative, which can reduce the computing time resulted in by the nonlocality of fractional derivative; Liu et al [25], Liu et al [27], and Yin et al [14] considered the fast calculation for time FPDEs based on the Xu' s two-grid FE methods [26], which can reduce the calculating time yielded by the nonlinear term; Zhao et al [30] developed a fast Hermite FE algorithm to improve the computational efficiency of Matrix, and presented a block circulant preconditioner; Yuste and Quintana-Murillo [28] presented the fast and robust adaptive methods with finite difference scheme for the time fractional diffusion equations; Xu et al [22], Wu and Zhou [23] considered the parareal algorithms for solving the linear time fractional ordinary or partial differential equations (FO(P)DEs), respectively; Zeng et al [8] presented a unified stable fast time-stepping method for fractional derivative and integral operators. Recently, Liu et al [9] proposed a fast TT-M FE algorithm for time fractional water wave model, which is developed to deal with time-consuming problem of nonlinear iteration used in the standard nonlinear Galerkin FE method for nonlinear term. In addition to the mentioned algorithms, there exist many other fast techniques to FPDEs or FODEs, which we will not list in this article.…”

“…Here, our work is to continue developing the new fast TT-M FE algorithm [9] to the nonlinear space fractional Allen-Cahn problems ∂u ∂t − ǫ 2 L α u + f (u) = g(z, t), (z, t) ∈ Ω × J, (1.1) with boundary condition u(z, t) = 0, (z, t) ∈ ∂Ω ×J, (1.2) and initial condition u(z, 0) = u 0 (z), z ∈ Ω, (1.3) where the coefficient ǫ is a given constant, α ∈ (1, 2) is the order of the fractional derivative, f (u) = u 3 − u is the nonlinear item, u 0 (z) is a given initial function, (1.5)…”

“…Here, our aim is to develop the fast TT-M FE algorithm proposed for solving the time FPDE [9] to solve nonlinear space fractional Allen-Cahn equations. The time direction is approximated by second-order θ scheme [10] derived based on the idea of the second-order α-schemes (See Galerkin FE method by Wang, Liu et al [11] in 2016; finite difference schemes by Gao et al [12] in 2015 and Sun et al [13] in 2016).…”