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Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems
Dongfang Li, Hong-lin Liao, Weiwei Sun, Jilu Wang, Jiwei Zhang
TL;DR
The paper addresses limited analysis of time-fractional nonlinear L1 methods, primarily due to the lack of a fundamental Gronwall-type inequality. It establishes an inequality for the L1 Caputo approximation and uses it to derive error estimates for fully discrete linearized L1-Galerkin FEMs, with applications to several model equations. The analysis includes multiple schemes with temporal orders 1 and 2−α, while accuracy can deteriorate without sufficient solution regularity.
Problem
Analysis of L1-type methods for time-fractional nonlinear problems is limited mainly because a fundamental Gronwall-type inequality is unavailable.
Method
The paper establishes a Gronwall-type inequality for the L1 approximation to the Caputo derivative and applies it to fully discrete linearized L1-Galerkin FEMs.
Results
The proposed schemes achieve temporal convergence orders 1 and 2−α, and the analysis provides optimal error estimates for the fully discrete methods.
Takeaways & Limitations
The analysis removes the local-time and decreasing-solution restrictions identified in previous works and is illustrated on linear Fokker-Planck, nonlinear Huxley, and Fisher equations.
Takeaways & Limitations
Without the requisite solution regularity, the truncation error may lose accuracy; an example also has an initial layer because its time derivative blows up near t = 0.
Abstract
from arXiv · showhide
This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of $L1$-Galerkin finite element methods. The analysis of $L1$ methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the $L1$ approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to three examples: linear Fokker-Planck equation, nonlinear Huxley equation and Fisher equation.
1 Introduction
The paper studies L1-Galerkin finite element methods for time-fractional nonlinear parabolic equations and addresses limited analysis caused by the lack of a fundamental Gronwall inequality. It establishes such an inequality and uses it to obtain error estimates for fully discrete methods.
- Problem setting: The model considers a time-fractional nonlinear parabolic equation with initial and homogeneous Dirichlet boundary conditions on a bounded convex polygonal domain.The domain may lie in R1, R2, or R3.
- Motivation: L1-type direct methods are widely used because they are easy to implement, but their numerical analysis remains limited even for linear models.The L1 scheme is a piecewise linear approximation to the fractional derivative.
- Motivation: Existing analyses of nonlinear L1 schemes often require local-in-time conditions, global Lipschitz assumptions, or restrictions related to decreasing numerical solutions.These conditions limit the scope of convergence and stability results.
- Motivation: For fractional orders 0 < α < 1, analysis is hindered mainly by the absence of a fundamental Gronwall-type inequality analogous to the classical parabolic case.The classical Gronwall inequality and its discrete counterpart are central to parabolic PDE analysis.
- Contribution: The paper establishes a new Gronwall-type inequality for L1 approximations and applies it to fully discrete L1-Galerkin FEMs for arbitrary prescribed T > 0.The resulting analysis covers linear or nonlinear source terms and several direct numerical methods.
- Contribution: The paper presents three linearized fully discrete schemes combining L1 time discretization with Galerkin finite elements in space, alongside convergence analysis and numerical experiments.The schemes and convergence results are developed in Sections 2 and 3, with experiments in Section 4.
2 L1-Galerkin FEMs and main results
The paper develops fully discrete linearized L1-Galerkin finite element methods for time-fractional nonlinear parabolic problems and derives their error estimates using a discrete Gronwall-type inequality. The analysis covers a basic scheme and two higher-order linearized schemes, while accounting for possible loss of accuracy from limited solution regularity.
- Methods: The methods combine an L1 approximation in time with Galerkin finite elements in space for nonlinear parabolic problems.The L1 approximation is based on piecewise linear interpolation of the Caputo fractional derivative.
- Basic scheme: Under sufficient regularity and a sufficiently small time step, the basic scheme has a unique finite element solution and an optimal error estimate.The estimate is expressed in terms of temporal and spatial discretization parameters, with the constant independent of τ and h.
- Regularity limitation: Limited regularity near t = 0 can reduce the temporal accuracy because the exact solution may develop an initial layer.The paper notes that smooth initial data and nonlinearities do not always guarantee the regularity required by the standard estimate.
- Error analysis: A discrete Gronwall-type inequality is the key analytical tool for proving the basic scheme’s error estimate.The same inequality also supports the estimate with reduced temporal accuracy when the exact solution lacks the requisite regularity.
- Higher-order schemes: Newton-linearized and extrapolated L1-Galerkin schemes achieve temporal convergence order 2−α, whereas the basic scheme has temporal order 1.These higher-order schemes are analyzed by extending the approach used for the basic linearized scheme.
3 Error analysis
The error analysis establishes a discrete Gronwall-type inequality for the L1 approximation and uses it to prove optimal error estimates for the fully discrete schemes. The proof combines projection and truncation-error estimates with sequence and matrix properties, under a sufficiently small time-step condition.
- Gronwall inequality: A new Gronwall-type inequality for nonnegative sequences is the central tool in proving the proposed schemes’ optimal error estimates.The inequality applies when the time step satisfies τ ≤ τ∗.
- Well-posedness: The coefficient matrix of the linearized FEM system is symmetric positive definite, so the numerical system has a unique solution.This establishes existence and uniqueness for the FEM system before completing the main error estimate.
- Error decomposition: The exact solution is compared with the numerical solution through Ritz projection, interpolation, truncation-error, and discrete-error equations.The analysis introduces the exact-solution equation, truncation error, and error equation after subtracting the numerical scheme.
- Error estimate: Combining the discrete estimates with the small-step condition completes the proof of the theorem’s optimal error bound.The final argument chooses τ0 ≤ τ∗ and a constant C0 sufficiently large.
- Gronwall proof: The Gronwall proof uses auxiliary functions, piecewise linear interpolation, weighted summation, and matrix representations of the sequence inequalities.The proof derives bounds through concavity arguments and properties of an upper triangular matrix.
4 Numerical examples
The numerical experiments test the proposed linearized L1-Galerkin schemes on Huxley, Fisher, and Fokker-Planck equations. Results support the predicted spatial and temporal convergence behavior under smooth solutions, while an initial singularity reduces the temporal rate for the Fokker-Planck example.
- Example 1: Huxley equation: For the Huxley equation, schemes (2.11) and (2.12) achieve temporal accuracy of order 2−α, while scheme (2.6) achieves order 1.The comparison uses quadratic finite elements and refined temporal meshes.
- Example 1: Huxley equation: For the Huxley equation, scheme (2.6) has optimal spatial convergence order r + 1.The result is reported for linear and quadratic finite elements with refined spatial meshes.
- Example 2: Fisher equation: For the Fisher equation, Tables 3 and 4 confirm the theoretical temporal and spatial convergence analysis of the three proposed schemes.The experiment uses quadratic finite elements, M = 60, and refined temporal meshes.
- Example 3: Fokker-Planck equation: For the Fokker-Planck equation, the exact solution has an initial layer because its time derivative blows up as t → 0+.The computation therefore tests temporal convergence without the regularity required for the standard rates.
- Example 3: Fokker-Planck equation: The three Fokker-Planck schemes converge, but their temporal rates are no longer order 1 or 2−α.The observed behavior agrees with the theoretical result stated in Remark 1.
5 Conclusions
The paper addresses limitations in L1-Galerkin analysis for time-fractional nonlinear parabolic problems by establishing a fundamental Gronwall type inequality. It uses that inequality to derive optimal error estimates without earlier local-time or decreasing-solution restrictions, and supports the analysis with numerical examples.
- The paper establishes a fundamental Gronwall type inequality for the L1 approximation to the Caputo fractional derivative.
- Using this inequality, the paper derives optimal error estimates for several fully discrete linearized L1-Galerkin finite element methods without previous restrictions.Earlier analyses generally required a small time interval or decreasing numerical solutions.
- Numerical examples across three models illustrate the theoretical results.