A two-grid finite element approximation for a nonlinear time-fractional Cable equation

Abstract

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Grünwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved; Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with O(τ 2−α + τ 2−β ). Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.

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

“…Different fast algorithms, which cover the fast computation of time fractional derivative, fast algorithm of nonlinear problem in time, fast calculation of nonlinear problem in space, fast computation of Matrix and so forth, have different acceleration strategies and features. 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.…”

TGMFE algorithm combined with some time second-order schemes for nonlinear fourth-order reaction diffusion system

“…Compared to the nonlinear Galerkin FE method, the two-grid method can save the CPU time and also get the almost the same errors and convergence rate to the one of nonlinear Galerkin FE method. In view of the advantages of two-grid method, the method has been developed by increasing researchers, the detailed contents can be found in Dawson and Wheeler [6], Chien and Jeng [9], Mu and Xu [8], Wu and Allen [13], Chen et al [10], Chen and Chen [12], Liu et al [14], Chen and Liu [15], Shi and Yang [7], Weng et al [16], Bajpai and Nataraj [18], Zhong et al [28], Liu et al [19], Liu et al [27], Yan et al [30] and some other references. Based on these discussions for two-grid method, ones can see that the time direction is approximated mainly by the second-order Crank-Nicolson (CN) scheme, the second-order two step backward difference (BD) method, and backward Euler (BE) method with first-order convergence rate.…”

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

“…The main idea of this method is using a coarse-grid space to produce a rough approximation of the solution for nonlinear problems, and then use it as the initial guess for one Newton-like iteration on the fine grid. Two-grid discretization method has been widely used for different kinds of problems, such as elliptic Equations [9,10], parabolic equations [12][13][14][15][16][17], eigenvalue problems [18][19][20] stochastic partial differential equations [21] and fractional differential equations [22,23]. The two-grid discretization idea is also used for nonlinear coupled equations, such as the complicated miscible displacement problems [24][25][26] and fluid flow in porous media [27].…”