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An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data
Bangti Jin, Raytcho Lazarov, Zhi Zhou
TL;DR
Existing L1 convergence theory relies on C2 time regularity that is generally unavailable for subdiffusion, including with smooth initial data. The paper revisits the analysis using data-regularity-based estimates and establishes first-order convergence for smooth and nonsmooth data, including more general sectorial and space-time fractional problems.
Problem
Existing O(τ^{2−α}) L1 analysis assumes C2 time regularity, while subdiffusion solutions generally have limited regularity and nonsmooth data are not covered.
Method
The paper develops data-regularity-based error analysis for fully discrete L1–Galerkin schemes and extends it to sectorial operators and one-dimensional space-time fractional problems.
Results
The analysis establishes first-order convergence for smooth and nonsmooth initial data, with numerical experiments confirming estimate sharpness and robustness.
Takeaways & Limitations
For any fixed tn > 0, the L1 scheme can achieve first-order convergence, and the analysis also provides error estimates for space-time fractional equations.
Takeaways & Limitations
Uniform first-order convergence on a nonuniform time mesh remains an open problem because the generating-function approach does not directly apply there.
Abstract
from arXiv · showhide
The subdiffusion equation with a Caputo fractional derivative of order $α\in(0,1)$ in time arises in a wide variety of practical applications, and it is often adopted to model anomalous subdiffusion processes in heterogeneous media. The L1 scheme is one of the most popular and successful numerical methods for discretizing the Caputo fractional derivative in time. The scheme was analyzed earlier independently by Lin and Xu (2007) and Sun and Wu (2006), and an $O(τ^{2-α})$ convergence rate was established, under the assumption that the solution is twice continuously differentiable in time. However, in view of the smoothing property of the subdiffusion equation, this regularity condition is restrictive, since it does not hold even for the homogeneous problem with a smooth initial data. In this work, we revisit the error analysis of the scheme, and establish an $O(τ)$ convergence rate for both smooth and nonsmooth initial data. The analysis is valid for more general sectorial operators. In particular, the L1 scheme is applied to one-dimensional space-time fractional diffusion equations, which involves also a Riemann-Liouville derivative of order $β\in(3/2,2)$ in space, and error estimates are provided for the fully discrete scheme. Numerical experiments are provided to verify the sharpness of the error estimates, and robustness of the scheme with respect to data regularity.
1. Introduction
The paper revisits L1 error analysis because existing O(τ^{2−α}) theory assumes unavailable C2 time regularity. It establishes data-regularity-based error bounds and reports first-order convergence for smooth and nonsmooth data.
- Motivation: Subdiffusion models anomalous transport, and Caputo fractional derivatives are used in applications involving heterogeneous media.Examples include fractal thermal media, heterogeneous aquifers, and underground environmental problems.
- Existing analysis: Existing L1 analyses derive an O(τ^{2−α}) local truncation bound under twice-continuous time differentiability.The L1 scheme is a widely used time discretization for the Caputo derivative.
- Problem: Limited smoothing makes the C2-regularity assumption restrictive, leaving nonsmooth data outside existing convergence analysis.The α-th order Caputo derivative can already be unbounded near the initial time.
- Problem: Numerical results indicate only first-order accuracy, even for smooth initial data, rather than the predicted O(τ^{2−α}) rate.This discrepancy motivates revisiting the L1 convergence analysis.
- Contribution: The paper establishes optimal error bounds expressed directly through data regularity and supports them with numerical experiments.The analysis covers both smooth and nonsmooth data and extends to more general sectorial operators.
- Contribution: For fixed tn > 0, the fully discrete scheme can achieve first-order convergence, although error estimates deteriorate as t approaches zero.The result applies to both v ∈ L2(Ω) and Av ∈ L2(Ω).
2. Preliminary
The preliminary section constructs spatially semidiscrete and fully discrete Galerkin–L1 schemes and derives solution representations used in the error analysis. Generating-function and contour representations provide the analytical foundation.
- Semidiscrete scheme: The spatial semidiscrete scheme uses a standard Galerkin finite element method, with a continuous piecewise linear space over a quasi-uniform triangulation.The construction introduces projection operators and a discrete Laplacian.
- Solution representations: Laplace-transform representations are derived for the continuous and semidiscrete homogeneous solutions.These representations use the analytic extension of the solution and the operator A = −∆ with homogeneous Dirichlet conditions.
- Fully discrete scheme: The fully discrete method combines L1 time stepping with Galerkin spatial discretization on a uniform time grid.The fully discrete solution is analyzed through a discrete analogue of the solution representation.
- Generating-function analysis: The discrete generating-function representation uses the L1 weights and the polylogarithm function.Contour deformation yields an alternative representation that forms the basis of the error analysis.
3. Error analysis of the fully discrete scheme
The analysis derives error estimates for the fully discrete L1 scheme using solution representations, contour arguments, kernel bounds, and sector-preserving properties. It establishes first-order convergence and L2 stability, while explaining the observed behavior for smooth and nonsmooth data.
- Error representation and bounds: The fully discrete error is analyzed by subtracting semidiscrete and fully discrete solution representations and separately bounding the resulting terms.The proof uses contour representations, resolvent estimates, and bounds on kernel differences.
- Kernel estimates: The kernel analysis establishes uniform contour bounds, including |χ(z) − z| ≤ c|z|^2τ and c1|z| ≤ |χ(z)| ≤ c2|z|.These estimates support comparison between the continuous and discrete operator kernels.
- Sectorial analysis: The mapping χ1 preserves sectors, enabling resolvent estimates for the discrete operator and control of the difference between B1(z) and B2(z).The sector-preserving property is identified as fundamental to the subsequent error analysis.
- Convergence and stability: The analysis establishes an O(τ) time-discretization error for nonsmooth initial data v ∈ L2(Ω), and the L1 scheme is L2(Ω)-stable.The stability conclusion follows directly from the nonsmooth-data error theorem.
- Convergence and stability: For smooth initial data, the L1 scheme has the same O(τ) convergence behavior as backward-Euler convolution quadrature, with a time-error singularity t^(α−1).The first-step expansion also exhibits a leading τ^α term for small τ.
4. Time-space fractional differential problem
The paper extends its sectorial-operator theory to one-dimensional space-time fractional diffusion involving a Riemann–Liouville spatial derivative. It constructs a Galerkin finite element discretization and provides fully discrete error estimates.
- Sectorial-operator extension: The extended theory applies to sectorial operators satisfying specified resolvent-sector and resolvent-bound conditions.The technical restriction on the sector angle is tied to the sector-preserving analysis.
- Model and regularity: The model combines a Caputo time derivative with a Riemann–Liouville spatial derivative to describe anomalous diffusion involving long-range interactions and history effects.The weak formulation uses a coercive and bounded sesquilinear form on the fractional Sobolev space.
- Model and regularity: The spatial operator’s domain includes a term x^(β−1), which indicates that solutions generally have limited regularity.This feature arises in the fractional spatial setting considered by the paper.
- Sectorial-operator extension: For β ∈ (3/2, 2), the Riemann–Liouville spatial operator has a resolvent set containing the required sector.The admissible angle satisfies θ ∈ ((2 − β)π/2, π/4).
- Fully discrete scheme: The fully discrete method combines L1 time stepping with a Galerkin finite element approximation on a uniform one-dimensional mesh.The finite element space consists of continuous piecewise linear functions, and the error estimates follow from the time and semidiscrete analyses.
5. Numerical experiments and discussions
Numerical experiments test the L1 scheme for smooth and nonsmooth data in subdiffusion and time-space fractional problems. They confirm first-order convergence and examine its dependence on fractional orders and proximity to t=0.
- Subdiffusion: O(τ) convergence is observed for smooth and nonsmooth subdiffusion data, independently of the fractional order α.The experiments use a fine spatial mesh so that temporal convergence dominates.
- Subdiffusion: The temporal error increases with α at fixed t, consistent with slower solution decay near t=0 for larger α.The temporal error also deteriorates as tn approaches zero according to the stated estimates.
- Subdiffusion: For α = 0.5, the error behaves like t1/2 for fixed N, and the scheme is not uniformly first-order in time even for very smooth initial data.The observed behavior confirms the singularity predicted for the time-discretization error.
- Time-space fractional problem: First-order convergence is observed for the time-space fractional problem, independently of the time- and space-fractional orders in the tested cases.The observation also holds for β = 5/4, outside the theory stated for the considered operator class.
- Time-space fractional problem: For nonsmooth time-space fractional data, first-order convergence remains robust at t = 0.1, 0.01, and 0.001.The experiments use β = 1.5 and examine increasingly small times.
- Time-space fractional problem: For fixed α, the time-space fractional error decreases as β increases, consistent with faster solution decay as β approaches two.Values of β closer to one correspond to more singular solutions and more challenging numerical approximation.
6. Conclusions
The paper establishes first-order convergence of the L1 scheme for smooth and nonsmooth initial data, extending the analysis beyond C2 time regularity. The framework also covers more general sectorial operators and space-time fractional equations, while identifying open issues for nonuniform meshes and diffusion-wave problems.
- First-order convergence is rigorously established for the L1 scheme with both smooth and nonsmooth initial data.
- The analysis applies to more general sectorial operators and includes error estimates for space-time fractional differential equations.
- Numerical experiments verify the sharpness of the error estimates and the scheme’s robustness with respect to data regularity.
- Because of solution singularity near t = 0, nonuniform meshes are a natural route toward uniform first-order convergence, but the presented generating-function approach does not directly apply there.
- Rigorous error estimates in terms of data regularity and convergence rates for diffusion-wave equations remain open problems.