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Error estimate of the nonuniform BDF3-L2 method for subdiffusion equations via multiscale solution decomposition
Wenlin Qiu, Kexin Li, Yiqun Li, Hao Zhang
TL;DR
The paper investigates why nonuniform L2 methods for subdiffusion do not consistently attain the predicted 3−α temporal order. It combines multiscale solution decomposition, spectral truncation, and a nonuniform BDF3–L2 method, proving temporal order 2+α and confirming it numerically.
Problem
Initial-time singularities reduce numerical accuracy, while direct multiscale decomposition imposes comparatively strong boundary conditions on the source term and initial data.
Method
The paper decomposes the solution to obtain a smoother unknown, applies spectral truncation to relax boundary constraints, and discretizes the truncated model using nonuniform BDF3–L2 and finite elements.
Results
The method has a rigorous temporal error estimate of order 2+α and numerical experiments observe order 2+α in time, with second-order spatial convergence.
Takeaways & Limitations
The MSD-based BDF3–L2 scheme achieves high-order temporal accuracy on uniform and graded meshes, including the uniform grade without the condition r > 3−α.
Takeaways & Limitations
The truncated formulation may require higher-order interpolation-based quadrature when the source term is structurally complex in time.
Abstract
from arXiv · showhide
Numerical experiments reported by Quan and Wu [SIAM J Numer Anal 61 (2023) 2106-2132] show that the observed temporal convergence rates of nonuniform L2 methods for subdiffusion models are not consistent with the theoretically predicted order $3-α$. This discrepancy suggests that a more refined analysis is needed and motivates the development of a nonuniform BDF3-L2 method for the subdiffusion equation. To account for the initial solution singularity, we employ the multiscale solution decomposition to decompose the original solution and approximate a smoother unknown variable that satisfies the subdiffusion model with a smoother source term. The resulting formulation, however, involves restrictive high-order boundary conditions on the source term and initial data. To overcome this difficulty, we introduce a spectral truncation technique that requires only slightly stronger regularity of the data and a controllable truncation error. We establish high-order regularity estimates of the solution to the truncated problem and develop a nonuniform BDF3-L2 method for its numerical approximation, based on which we derive a rigorous error estimate of temporal convergence order $2+α$. Numerical experiments are carried out to substantiate the theoretical findings.
1 Introduction
The paper addresses reduced accuracy caused by initial-time solution singularities in nonuniform L2 methods for subdiffusion. It combines multiscale solution decomposition with spectral truncation and a BDF3–L2 discretization to support higher-order analysis.
- 1.1 Main motivation: Subdiffusion solutions exhibit weak initial-time singularities that significantly reduce numerical approximation accuracy.Graded and nonuniform meshes, correction-based convolution quadratures, and L2-type schemes are among existing approaches for recovering convergence.
- 1.1 Main motivation: The multiscale solution decomposition separates the singular solution into known singular terms and a smoother unknown with smoother forcing.The smoother variable still satisfies the original equation, allowing numerical analysis based on solution smoothness.
- 1.2 Challenges and contributions: Applying MSD directly introduces comparatively strong high-order boundary conditions on the source term and initial data.These conditions require Δ^k(f + Δu0) = 0 on the boundary for 0 ≤ k ≤ m − 1, together with an L2 condition on Δ^m(f + Δu0).
- 1.2 Challenges and contributions: Spectral truncation relaxes these boundary constraints while requiring only slightly stronger data regularity and producing a controllable truncation error.The truncation uses the eigenfunction expansion of the Dirichlet Laplacian.
- 1.2 Challenges and contributions: The paper proves high-order regularity for the transformed problem and develops time-semidiscrete and fully discrete BDF3–L2 finite element schemes.The stability and optimal fully discrete error estimates are established, with numerical examples used to substantiate the theory.
2 Solution regularity for truncated model
The truncated-model analysis establishes high-order regularity for the smoother solution and quantifies the effect of spectral truncation. The truncation error can be reduced by increasing the spectral cutoff, while complex source terms may require higher-order quadrature.
- 2.1 Regularity estimates: The paper derives regularity estimates for the smoother variable in the transformed subdiffusion model.The analysis differentiates the solution representation and bounds spatial and temporal derivatives using Mittag-Leffler-function estimates.
- 2.1 Regularity estimates: High-order regularity estimates are established under assumptions on the initial data and time derivatives of the source term.Theorem 2.1 assumes bounded ∥Δ^(m+2)u0∥ and ∥Δ^(m+1)∂_t f(·,t)∥ ≤ Q t^−σ.
- 2.2 Truncated model: Spectral truncation replaces the highest-order source component by its eigenfunction projection, automatically satisfying the required homogeneous boundary conditions.The truncated model therefore relaxes the comparatively strong constraints imposed on the untruncated formulation.
- 2.2 Truncated model: The truncation contributes an additional error of order Qλ_(L+1)^−γ̂/2 ∥f̃∥_{C([0,T];H^(2m+γ̂)(Ω))}.This error can be made arbitrarily small by choosing a sufficiently large cutoff L.
- 2.2 Truncated model: For temporally complex source terms, higher-order interpolation-based quadrature can approximate the right-hand-side integrals without degrading overall accuracy.The remark identifies the L2 method as an example of such a quadrature.
3 Stability of the time-semidiscrete scheme
The time-semidiscrete scheme combines nonuniform L2 discretization of the fractional derivative with variable-step BDF3 time differentiation. Stability is proved under explicit step-ratio restrictions.
- 3.1 Construction of the time-semidiscrete scheme: The scheme uses the L2 method for the Caputo derivative on a general nonuniform time mesh.Linear and quadratic Lagrange interpolation operators are used on the relevant time intervals.
- 3.2 Stability analysis: The stability analysis relies on step-ratio conditions ensuring the required discrete inequalities.Lemma 3.1 uses ρ_k ∈ [ρ_L, ρ_R], while the main stability theorem assumes ρ_k ∈ [0.5, 1.7319].
- 3.1 Construction of the time-semidiscrete scheme: The first two time steps use backward Euler, while later steps use a variable-step BDF3 approximation for the time derivative.The construction treats k = 1, 2 separately from k ≥ 3.
- 3.2 Stability analysis: The resulting time-discrete scheme approximates the truncated model together with the relation recovering the original solution.The numerical variables replace the truncated continuous variables after truncation residuals are omitted.
- 3.2 Stability analysis: Theorem 3.3 establishes stability of the time-discrete scheme under the stated step-ratio restriction.The theorem applies to solutions of the scheme defined by equations (34)–(36).
4 Error estimates
The section derives time-discrete and fully discrete Galerkin error estimates for the truncated problem under bounded-data and mesh-ratio assumptions, obtaining temporal order 2+α and spatial approximation control.
- Time-discrete error estimate: The analysis establishes auxiliary lemmas and regularity bounds under assumptions on the initial data, source-term derivatives, mesh ratios, and spatial regularity.These assumptions include ρ_k ∈ [ρ_L, ρ_R], m ≥ 3, and a source bound of the form ∥∆^{m+1}∂_t^3 f(·,t)∥ ≤ Q t^{-σ}, with 0 < σ < 1.
- Time-discrete error estimate: For ρ_k ∈ [1, 1.7319] and m ≥ 3, Theorem 4.3 gives the time-discrete error estimate for the numerical variables.The proof introduces error equations by subtracting the numerical scheme from the corresponding truncated-problem relations and estimates the residual terms.
- Time-discrete error estimate: The time-discrete temporal error is bounded with convergence order 2+α through a sum involving the time-step powers τ_k^(2+α).The estimate is obtained by combining residual bounds with the stability result used in the proof.
- Fully discrete Galerkin error estimate: The fully discrete scheme uses continuous piecewise linear finite elements on a quasi-uniform spatial partition with mesh diameter h and a Ritz projection I_h.The Galerkin formulation is constructed for the truncated problem after omitting the local truncation errors in the time-discrete scheme.
- Fully discrete Galerkin error estimate: If ρ_k ∈ [0.5, 1.7319], Corollary 4.4 gives a stability and convergence estimate for the fully discrete Galerkin solution.Theorem 4.5 then combines this framework with approximation properties to establish the fully discrete error estimate under the assumptions of Theorem 4.3.
- Fully discrete Galerkin error estimate: The error for the original solution is obtained by combining the fully discrete error with the spectral truncation error through the triangle inequality.Because the truncation error tends to zero as L → ∞, L can be chosen sufficiently large to control this additional contribution.
5 Numerical experiments
Numerical experiments test MSD-based BDF3–L2 schemes across one- and two-dimensional examples, showing the predicted temporal and spatial convergence behavior. They also show that increasing the MSD scale improves solution smoothness and numerical accuracy.
- Experimental setup: The experiments use graded temporal meshes, two-mesh discrete L2 errors, and fixed parameters T = 1, J = 64, and N = 32.The graded-mesh ratio is selected within the range required by Theorem 4.3.
- Effects of MSD on numerical accuracy: Increasing m improves the smoothness of Vh by improving the regularity of the right-hand side, especially near the initial singularity.For m = 0, both Vh and Uh remain weakly singular near the initial time.
- Effects of MSD on numerical accuracy: Improved regularity of Vh enhances the numerical accuracy of Uh as an approximation to u.The experiments use BDF3–L2 approximations at x = π/2 to illustrate this MSD effect.
- Comparison of convergence behavior: 2+α temporal accuracy is achieved by the MSD-based BDF3–L2 scheme on a uniform mesh, whereas the compared L2 scheme achieves only first-order accuracy.The alternative order 3−α is reported only under the condition r > 3−α and is associated with greater numerical difficulties.
- Comparison of convergence behavior: 2+α temporal and second-order spatial convergence are observed for the BDF3–L2 scheme on both uniform and graded meshes in Example 3.The observations are reported as consistent with Theorem 4.5.
- Convergence behavior of the scheme (45): 2+α temporal and second-order spatial accuracy are again observed for the fully discrete scheme in the two-dimensional experiments.Tables 5–6 report these convergence behaviors for Example 4.