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Keywords:Time-fractional fourth-order reaction-diffusion problem Finite element method Finite difference scheme Caputo-fractional derivative Unconditional stability Optimal a priori error analysis a b s t r a c tIn this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction-diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ = ∆u. Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L 2 -norm with optimal order of convergence O(∆ 2−α t +h m+1 ), where ∆ t and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.
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.
In this article, an H 1 -Galerkin mixed finite element (MFE) method for solving time fractional reaction-diffusion equation is presented. The optimal time convergence order O(Δt 2−α ) and the optimal spatial rate of convergence in H 1 and L 2 -norms for variable u and its gradient σ are derived. Moreover, some numerical results are shown to support our theoretical analysis.
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