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Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions
Natalia Kopteva
TL;DR
The paper addresses error analysis for L1-type discretizations of a Caputo fractional parabolic problem with initial-time solution singularities. It develops a framework covering graded and uniform temporal meshes, finite difference and finite element spatial discretizations, and L2 and L∞ norms. For the model problem, graded meshes achieve M^−min{αr,2−α}, while the singular uniform-mesh case has an M^−α error bound.
Problem
Error analysis is needed for L1-type discretizations of fractional parabolic problems whose solutions typically exhibit singular behaviour at the initial time.
Method
The paper develops a simple framework based on temporal-error estimates and stability properties for L1-type schemes on graded and uniform meshes, with finite difference and finite element spatial discretizations.
Results
For the model problem, graded meshes yield |u(t_m)−U^m| ≲ M^−min{αr,2−α}, while uniform meshes under the singularity assumption yield an error bounded by M^−α.
Takeaways & Limitations
The framework supplies error estimates across graded and uniform temporal meshes and extends to both finite difference and finite element spatial discretizations in L2 and L∞ norms.
Takeaways & Limitations
The main regularity assumption allows initial-time singularities but excludes a less-singular case in which the initial condition is uniquely determined by the other problem data.
Abstract
from arXiv · showhide
An initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.
1. Introduction
The paper develops a concise error-analysis framework for L1-type discretizations of a Caputo fractional parabolic problem, covering graded and uniform meshes, finite difference and finite element spatial schemes, and L2 and L∞ norms.
- Problem setting: The problem is posed on a bounded Lipschitz domain in dimensions d ∈ {1, 2, 3}, with a Caputo derivative of order α ∈ (0, 1) and a linear second-order elliptic spatial operator.The problem includes homogeneous boundary data and an initial condition.
- Problem setting: The analysis assumes a unique solution whose temporal derivatives may be singular near t = 0, reflecting typical solution behaviour.The authors describe this assumption as more realistic than the stronger bounded-derivative assumptions often used in the literature.
- Scope: The assumed singularity excludes a less-singular case in which the initial condition would be uniquely determined by the other problem data, but the results also apply to nonsingular or differently singular solutions.The authors identify the excluded implication as too restrictive.
- Framework: The framework analyzes L1-type schemes on graded and uniform temporal meshes and applies to both finite difference and finite element spatial discretizations.It targets errors in both L2(Ω) and L∞(Ω) norms.
- Framework: The framework extends to time-dependent elliptic operators and quasi-uniform or quasi-graded meshes, while recovering some previously known error bounds and establishing new ones.The paper presents the approach as simple and broadly applicable across the stated discretization settings.
- Analysis strategy: The paper introduces concise temporal-truncation-error representations and a shorter stability proof based on a barrier function for uniform meshes.The temporal-error analysis is first developed for a problem without spatial derivatives, then extended to semidiscrete and fully discrete schemes.
2. Paradigm for the temporal-discretization error analysis
The paper builds temporal-error estimates from graded-mesh stability and truncation-error representations, then adapts the framework to uniform meshes. It obtains rates M^−min{αr,2−α} on graded meshes and M^−α on uniform meshes for the singular model problem.
- 2.1. Graded temporal mesh: The graded temporal mesh is defined by t_j = T(j/M)^r, with r ≥ 1; r = 1 gives a uniform mesh, while larger r clusters points near t = 0.Quasi-graded meshes are also covered.
- 2.2. Stability properties: The discrete fractional operator satisfies stability bounds that control V^m by weighted maxima of the forcing sequence, providing the main mechanism for error propagation.A separate, subtler stability property is used for uniform meshes.
- 2.3. Error estimation: The temporal truncation error is represented through integral expressions involving the piecewise-linear interpolant, enabling estimates for the simplest fractional-derivative problem.These representations are combined with stability lemmas to bound the solution error.
- 2.3. Error estimation: M^−min{αr,2−α} bounds the error |u(t_m)−U^m| for the graded mesh under the stated derivative assumptions.The bound holds for m = 1, …, M.
- 2.3. Error estimation: r = (2−α)/α attains the optimal graded-mesh rate O(M^−(2−α)); larger r preserves the rate but increases mesh widths near t = T.The larger final-step widths can lead to larger errors.
- 2.4. Analysis on the uniform mesh: M^−α bounds the uniform-mesh error under the singularity assumption |∂_t^l u(t)| ≲ 1 + t^{α−l} for l = 1, 2.The estimate follows from the refined uniform-mesh stability property.
3. Error analysis for the L1 semidiscretization in time
The L1 time-semidiscretization error is bounded through temporal truncation errors, with estimates applying in Lp norms and sharper forms under additional regularity. The analysis also accommodates general uniformly elliptic spatial operators, although some fully discrete maximum-norm conditions may be problematic.
- General semidiscrete error bound: The semidiscrete error satisfies ∥u(·, t_m) − U^m∥_{L_p(Ω)} ≲ max_{j=1,...,m} ∥ψ_j∥_{L_p(Ω)}.This estimate holds for m = 1, ..., M under the stated condition.
- Sharper estimate for r = 1: For r = 1, the bound can be sharpened to max_{j=1,...,m}{τ^−γ t_j^{1−α+γ}∥ψ_j∥_{L_p(Ω)}}, with τ = T M^−1 and γ = min{α, 1 − α}.The sharper estimate replaces the right-hand side of the general error bound.
- Rates under solution regularity: Under the assumed time-derivative regularity, the semidiscrete error is bounded by M^−min{αr, 2−α}.The estimate holds for m = 1, ..., M.
- Rates under solution regularity: For r = 1, the corresponding bound becomes ∥u(·, t_m) − U^m∥_{L_p(Ω)} ≲ t_m^{α−1}.This sharper form is stated in addition to the M-dependent estimate.
- General elliptic operators: The framework extends to general uniformly elliptic operators while retaining the L2 energy argument and the L∞ maximum-principle estimate.For fully discrete finite elements, the maximum-norm condition A_p may be problematic; analogous finite-difference schemes are not readily available in the general case.
4. Maximum norm error analysis for finite difference discretizations
The finite-difference analysis applies the L1 method on tensor-product uniform spatial meshes and bounds the nodal maximum-norm error through temporal truncation and spatial discretization estimates. Under the stated assumptions, the resulting rates combine the temporal rate with an O(h^2) spatial term.
- Spatial discretization: The problem is posed on Ω = (0, 1)^d and discretized spatially using tensor products of d uniform meshes.The finite-difference error is measured in the nodal maximum norm ∥·∥_{∞;Ω_h}.
- Well-posedness: The discrete solution is unique under the stated assumptions and condition A_p.The assumptions include c ≥ 0 and the specified regularity and mesh setting.
- Temporal error: For r = 1, the temporal estimate can use max_{j=1,...,m}{τ^−γ t_j^{1−α+γ}∥ψ_j∥_{L∞(Ω)}}, with τ = T M^−1 and γ = min{α, 1 − α}.This is the sharper alternative to the general maximum-norm temporal bound.
- Combined error rate: The spatial truncation contributes O(h^2) under the stated spatial derivative assumptions.The proof combines temporal truncation bounds with the estimate |(L_h − L)u| ≲ h^2.
5. Error analysis for finite element discretizations
The finite-element analysis combines temporal semidiscretization with spatial Ritz projection estimates under norm- and mesh-dependent assumptions. It derives L2 and L∞ error bounds for linear and higher-degree elements, including graded-mesh temporal rates and spatial discretization terms.
- Discretization and assumptions: The spatial method applies standard finite elements on quasiuniform simplicial triangulations and analyzes the error through a Ritz projection.The discretization is posed on bounded Lipschitz domains in dimensions two or three.
- Discretization and assumptions: For L2(Ω) analysis, A2 requires the discrete inner product to equal the exact L2(Ω) inner product, while a quadrature variant is also justified under additional conditions.The quadrature-based norm is equivalent to the L2(Ω) norm on the finite-element space.
- Discretization and assumptions: For L∞(Ω) analysis, linear elements and non-positive off-diagonal stiffness-matrix entries ensure the required A∞ condition.The resulting stiffness matrix is an M-matrix, while the lumped mass matrix is positive diagonal.
- L2(Ω) error: M^-min{αr,2−α} + h^(ℓ+1) bounds the L2(Ω) error for m = 1, . . . , M under the stated p = 2 conditions.For r = 1, the corresponding temporal term is M^-1.
- L∞(Ω) error: M^-min{αr,2−α} + h^2|ln h| bounds the L∞(Ω) error for lumped-mass linear finite elements under the stated p = ∞ conditions.For r = 1, the temporal term becomes M^-1, while the spatial contribution remains h^2|ln h|.
- Scope and limitations: The triangulation condition for the L∞(Ω) result is restrictive but is satisfied by mildly structured meshes whose elements are close to equilateral triangles or regular tetrahedra.Non-obtuse triangulations are sufficient but not necessary for the condition.
6. Estimation of derivatives of the exact solution u
The section argues that the derivative assumptions used in the error analysis are realistic and develops eigenfunction-based arguments to verify them across spatial settings. It also identifies additional spatial-regularity constraints, especially on domains with corners.
- Temporal derivatives: Separation of variables uses eigenvalues, eigenfunctions, and Mittag-Leffler functions to represent the solution and analyze its temporal derivatives.The approach applies most directly to self-adjoint operators and has been used for smooth, polygonal, and polyhedral domains.
- Temporal derivatives: The bounds ∥∂_t^l u(·,t)∥_Lp(Ω) ≲ 1 + t^(α−l), for l = 1, 2 and p ∈ {2,∞}, are stated to be realistic.For d = 1, they follow under regularity and compatibility assumptions on u_0 and f, although the eigenfunction proof does not directly extend to d > 1.
- Temporal derivatives: For higher dimensions, applying L^q to the eigenfunction expansion establishes L_2 derivative bounds that imply the desired L_2 and L_∞ estimates.This modification relies on regularity assumptions on u_0 and temporal derivatives of f.
- Spatial derivatives: Additional assumptions on spatial derivatives are more delicate because corners in Ω can produce corner singularities.The issue affects the derivative hypotheses used for the finite difference and finite element error bounds.
- Spatial derivatives: For a two-dimensional example, elliptic regularity and eigenfunction estimates yield ∥u∥_W^4_∞(Ω) ≲ 1, satisfying the spatial assumptions of Corollary 4.2.The result follows under stated regularity and compatibility assumptions on u_0 and f.
- Finite element assumptions: For linear finite elements, temporal derivative bounds follow from the L_2 estimates, while higher-order elements require additional data regularity and smooth domains.A related example verifies the assumptions of Corollary 5.7 under regularity conditions on u_0 and f.
7. Numerical results
Numerical experiments test the finite element method on a two-dimensional problem with a known singular-in-time exact solution. The results confirm the predicted sharp convergence behavior on an optimally graded temporal mesh.
- Problem setup: The experiment uses a two-dimensional domain with nonhomogeneous boundary data chosen so that the exact solution is u = t^α cos(xy).The problem is discretized with lumped-mass linear finite elements on quasiuniform Delaunay triangulations.
- Temporal mesh: The graded temporal mesh is t_j = T(j/M)^r with the optimal grading parameter r = (2−α)/α.Errors are examined for a large fixed M and fixed degrees of freedom, with computational rates reported in the latter case.
- Results: The observed errors confirm the sharpness of the bound M^−(2−α) + h^2|ln h| predicted by Corollary 5.7(i).The comparison uses maximum nodal errors and computational convergence rates.
Appendix A. Proof of Lemma 2.1∗
The appendix proves a temporal truncation-error lemma by constructing barrier functions and estimating integral representations across graded-mesh index ranges. The proof combines local derivative estimates with a sufficiently large auxiliary parameter.
- Barrier construction: The proof introduces a barrier function B(s) and its mesh values B_j to estimate the truncation error for singular temporal behavior.The construction uses β = 1−α and tracks dependence on the auxiliary parameter p.
- Integral estimates: For later mesh intervals, the analysis compares tB(t) with tB(t_m) and rescales the integration variable by t.The identity α + β = 1 is used in the resulting estimates.
- Integral estimates: The proof estimates differences between discrete and continuous derivatives of auxiliary functions using bounds on F′′ and B′′ over separate index ranges.These estimates distinguish intervals before and after a threshold index n.
- Conclusion: Choosing p sufficiently large after combining the estimates yields the desired assertion for δ_m^α.The remaining cases are handled by introducing p_m and c_m and combining the resulting bounds recursively.
Appendix B. Lumped-mass quadrature error in the maximum norm
The appendix bounds the lumped-mass quadrature contribution to the Ritz-projection error in the maximum norm. Separate two- and three-dimensional arguments use discrete Sobolev or Green-function estimates.
- Error component: Lumped-mass quadrature adds a component ρ̂_h to the Ritz-projection error, characterized variationally through the difference between lumped and exact inner products.The appendix seeks a maximum-norm estimate for this component.
- Two dimensions: In two dimensions, the proof uses the discrete Sobolev inequality ∥ρ̂_h∥_L∞(Ω) ≲ |ln h|^1/2∥∇ρ̂_h∥_L2(Ω).Testing the variational relation with ρ̂_h produces the required estimate.
- Three dimensions: In three dimensions, the proof evaluates the maximum at an interior node and tests with a discrete Green’s function associated with that node.Bounds on the Green’s function and its gradient then yield the desired estimate.