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A Two-Grid Finite Element Approximation for A Nonlinear Time-Fractional Cable Equation
Yang Liu, Yanwei Du, Hong Li, Jinfeng Wang
TL;DR
The paper addresses numerical solution and analysis of the nonlinear time-fractional Cable equation. It develops a two-grid finite-element method with higher-order time approximation, and reports stable second-order temporal convergence independent of the fractional parameters, alongside lower computation time than standard nonlinear finite elements.
Problem
The paper addresses numerical solution and stability and error analysis for the nonlinear time-fractional Cable equation using a higher-order two-grid finite-element method.
Source: §1 · pp. 2–3Method
The method combines a two-grid finite-element algorithm with second-order time approximations, including a WSGD scheme for fractional derivatives.
Source: §1 · pp. 3–4, §2.2 · p. 6Results
The method is reported to achieve stable second-order temporal convergence independent of α and β and similar accuracy with less computation time than standard nonlinear finite elements.
Source: §6 · p. 17Takeaways & Limitations
The authors identify analysis of two-grid finite-element methods for nonlinear space and space-time fractional PDEs as future work.
Source: §6 · p. 17The analysis assumes a bounded convex polygonal domain in dimension d ≤ 2 and a nonlinear term F with bounded growth and derivative.
Source: §1 · p. 2
Abstract
from arXiv · showhide
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.
1 Introduction
The introduction situates the fractional Cable equation as a model of anomalous electro-diffusion in nerve cells and reviews existing numerical approaches. It identifies a gap in finite-element theory for WSGD-based two-grid methods and states the paper’s stability and error-analysis contributions.
1 Introduction
- Finite-element methods have been developed for many fractional PDEs, including time-fractional problems, with reported convergence and error analyses.Examples include a fourth-order time-FPDE method with first-order temporal convergence and time-FPDE methods with optimal error estimates.Source: §1 · p. 2
- The fractional Cable equation models anomalous electro-diffusion in nerve cells, and prior work has applied finite differences, spline collocation, spectral approximations, and finite elements to it.The cited finite-element studies use Riemann–Liouville derivatives and different fractional-derivative approximations.Source: §1 · pp. 2–3
- Two-grid methods have been used for integer-order PDEs and a Caputo fractional problem, but the latter used L1 approximation and achieved temporal order 2−α.The introduction states that two-grid finite-element analysis using WSGD had not been reported.Source: §1 · pp. 3–4
- The paper studies a two-grid finite-element method with WSGD for the nonlinear fractional Cable equation, deriving stability and a priori estimates with second-order temporal convergence.The stated temporal rate is higher than the L1-approximation rate O(τ^(2−α) + τ^(2−β)); the paper also compares computation time with standard nonlinear finite elements.Source: §1 · pp. 3–4
2 Fractional derivatives and two-grid FE method
The section introduces fractional-derivative definitions and a second-order time approximation, then constructs a two-grid finite-element scheme using coarse-grid nonlinear and fine-grid linear solves.
2.1 Fractional derivatives and approximate formula
- The time discretization uses a second-order fractional-derivative approximation and a two-step backward Euler approximation for the integer derivative.The time grid is defined by t_i = iτ.Source: §2.1 · p. 5, §2.2 · p. 6
- For zero initial value, the second-order discrete formula for the Riemann–Liouville derivative also approximates the Caputo derivative.The section relates the Caputo and Riemann–Liouville derivatives and supplies lemmas used in the subsequent analysis.Source: §2.1 · pp. 4–6
2.2 Two-grid algorithm based on FE scheme
- The two-grid scheme solves a nonlinear finite-element system on the coarse grid, then uses its solution to form a linear system on the fine grid.The authors describe this system as more efficient than the standard nonlinear finite-element system.Source: §2.2 · pp. 7–8
3 Analysis of stability based on two-grid FE algorithm
The section establishes a stability inequality for the fully discrete two-grid finite-element system.
3 Analysis of stability based on two-grid FE algorithm
- Theorem 1 establishes a stable inequality for the fine-grid solution of the two-grid finite-element system.The proof combines estimates for the fine-grid and coarse-grid systems.Source: §3 · pp. 8, 10
4 Error analysis based on two-grid algorithm
The error analysis uses a Ritz projection to derive a priori estimates for the fully discrete two-grid method. It gives a second-order temporal convergence rate independent of the fractional parameters.
4 Error analysis based on two-grid algorithm
- The error is decomposed into a projection component and a discrete-solution component.The section introduces the Ritz projection before deriving the error estimates.Source: §4 · p. 10
- The L2 error is bounded by C[τ² + (1 + τ^-α)h^(r+1) + (1 + τ^-2α)H^(2r+2)].The constant is independent of the coarse-grid size H, fine-grid size h, and time step τ.Source: §4 · p. 11
- The temporal convergence rate is second order, independent of α and β, and higher than the L1-approximation rate O(τ^(2−α) + τ^(2−β)).The analysis derives a priori errors using a Ritz-projection operator.Source: §4 · pp. 10, 15
5 Numerical Tests
The numerical tests compare the two-grid FE method with standard nonlinear FE and report similar accuracy with lower computational time. They also report stable second-order time convergence independent of the fractional parameters, exceeding the stated L1-approximation rate.
5 Numerical Tests
- The numerical example uses F(u) = u³ − u and exact solution u(x, y, t) = t² sin(2πx) sin(2πy) on [0, 1]² × [0, 1].It uses rectangular meshes, a two-step backward method for the first-order integer derivative, a second-order fractional scheme, and a continuous bilinear FE space.Source: §5 · pp. 15–16
- The tests report L2 errors and convergence orders for α = 0.01, β = 0.99; α = 0.5, β = 0.5; and α = 0.99, β = 0.01, with τ = 1/100.Table 1 covers α = 0.01 and β = 0.99; the reported comparisons use two-grid coarse and fine meshes and standard FE meshes.Source: §5 · p. 16, Table 1 · p. 17
- For α = 0.99, β = 0.01, τ = 1/100, and h = H² = 1/25, the two-grid and standard FE errors show similar computational accuracy.The figures compare surfaces for the exact solution and both FE solutions, as well as their errors.Source: §5 · p. 16
- The two-grid FE method has similar accuracy but lower computational time than standard nonlinear FE, with stable second-order convergence independent of α and β and higher than the stated L1 rate.The reported L1-approximation rate is O(τ^(2−α) + τ^(2−β)).Source: §5 · p. 16
6 Some concluding remarks
The paper develops a two-grid FE method for nonlinear fractional Cable equations and analyzes its stability and time-discretization error. Its numerical results report lower computational time than standard FE and second-order convergence independent of α and β.
6 Some concluding remarks
- The paper derives stability and a priori error estimates for its two-grid FE numerical scheme, with second-order convergence in time.The authors also compute numerical errors and convergence orders to verify the theoretical results.Source: §6 · p. 17
- The concluding results report that two-grid FE spends less time than FE and achieves second-order convergence independent of α and β, compared with the stated L1 rate O(τ^(2−α) + τ^(2−β)).The authors also report that the method solves the nonlinear time-fractional Cable equation well.Source: §6 · p. 17
- The authors identify numerical theory for two-grid FE methods on nonlinear space- and space-time-fractional PDEs as future work.Source: §6 · p. 17