A Two-Grid Finite Element Approximation for A Nonlinear Time-Fractional Cable Equation

TL;DR

This paper introduces a two-grid finite element method for solving a nonlinear time-fractional Cable equation, achieving higher temporal accuracy and reduced computational time through a combined coarse and fine grid approach.

Contribution

The paper presents a novel two-grid finite element scheme with second-order temporal accuracy for nonlinear fractional PDEs, improving efficiency and convergence over existing methods.

Findings

The proposed method achieves second-order convergence in time.

Numerical results confirm the efficiency and accuracy of the two-grid scheme.

CPU time is significantly reduced compared to traditional methods.

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 Gr\"unwald 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 $O(\tau^{2-\alpha}+\tau^{2-\beta})$. 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.