The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

Bangti Jin; Raytcho Lazarov; Yikan Liu; Zhi Zhou

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The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

Bangti Jin, Raytcho Lazarov, Yikan Liu, Zhi Zhou

arXiv:1401.8049v1Jan 31, 2014math.NA

TL;DR

The paper addresses error analysis for numerical schemes for multi-term time-fractional diffusion, especially with low-regularity data. It develops Galerkin finite element discretizations and derives nearly optimal semidiscrete estimates, while numerical results confirm predicted convergence rates.

Problem
Existing multi-term schemes assume sufficiently smooth solutions, leaving error analysis for low-regularity solutions as the main unresolved goal.
Method
The paper uses a Galerkin finite element approximation with continuous piecewise linear functions, combined with a finite-difference time discretization for the fully discrete scheme.
Results
The semidiscrete scheme achieves nearly optimal error estimates with respect to data regularity, while the fully discrete scheme has convergence rate O(h2 + τ 2−α) for smooth solutions.
Takeaways & Limitations
The analysis covers nonsmooth initial data and right-hand sides, and numerical experiments confirm the theoretically predicted convergence rates.
Takeaways & Limitations
For less regular source terms, a legitimate weak solution requires f ∈Lr(0, T; ˙Hq(Ω)) with r > 1/α and −1 < q ≤1; the solution may not satisfy u(x, 0) = 0.

Abstract

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We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.

1. introduction

The paper develops and analyzes Galerkin finite element schemes for multi-term time-fractional diffusion, targeting nearly optimal rates that remain valid for nonsmooth data. It also establishes a fully discrete finite-difference-in-time scheme and verifies the predicted convergence numerically.

Model and motivation: Multi-term fractional diffusion models extend single-term models used to describe anomalous diffusion in heterogeneous aquifers, viscoelastic materials, and transport with mobile and immobile solute states.The model contains multiple Caputo fractional derivatives with distinct orders and positive coefficients.
Research gap: Prior multi-term numerical analyses assumed sufficiently smooth solutions, leaving low-regularity cases insufficiently addressed.The paper identifies this gap as the main goal of the study.
Semidiscrete method: The semidiscrete method uses continuous piecewise linear finite element functions on shape-regular, quasi-uniform meshes to approximate the spatial problem.The discrete solution satisfies the Galerkin variational equation with an approximation of the initial data.
Error analysis: Nearly optimal semidiscrete error estimates cover homogeneous and inhomogeneous problems with both smooth and nonsmooth initial data and right-hand sides.The analysis includes initial data with −1 < q ≤ 2 and right-hand sides with −1 < q < 1 in the stated Sobolev scales.
Fully discrete method: O(h2 + τ 2−α) is the convergence rate established for the fully discrete scheme when the solution is smooth.The fully discrete method uses finite differences for the Caputo fractional derivatives and includes stability and error analysis.

2. Solution theory

The solution theory represents the multi-term fractional diffusion solution using multinomial Mittag-Leffler operators and establishes regularity for homogeneous and inhomogeneous problems, including nonsmooth data.

Solution representation: The multinomial Mittag-Leffler function generalizes the exponential function and appears in the solution representation of the multi-term model.When m = 1 and β = β1 = 1, it reproduces ez.
Solution representation: The solution operators E(t) and ¯E(t) represent the solution, while ¯E provides a smoothing property used in the regularity analysis.The operators are constructed through spectral separation and the Laplace eigenvalues and eigenfunctions.
Homogeneous problems: For homogeneous problems with v ∈ ˙Hq(Ω), −1 < q ≤2, the analysis derives regularity estimates covering gains up to two spatial derivatives.The corresponding ranges depend on the operator derivative index ℓ and the difference p − q.
Inhomogeneous problems: For f ∈L∞(0, T; ˙Hq(Ω)), −1 < q ≤1, and v = 0, the solution belongs to L∞(0, T; ˙Hq+2−ϵ(Ω)) for any ϵ > 0.The result provides spatial regularity for an inhomogeneous problem with less regular source terms.
Inhomogeneous problems: The regularity theory extends to f ∈L2(0, T; ˙Hq(Ω)), −1 < q ≤1, yielding u ∈L2(0, T; ˙Hq+2(Ω)); for α ∈(1/2, 1), this gives a legitimate weak solution.More generally, the source condition f ∈Lr(0, T; ˙Hq(Ω)) with r > 1/α preserves the initial condition in the stated sense.

3. Error Estimates for Semidiscrete Galerkin Scheme

The semidiscrete scheme uses continuous piecewise linear Galerkin finite elements, projection-based error splittings, and separate analyses for smooth and nonsmooth data.

Semidiscrete scheme: The analysis uses L2 and Ritz projections, their approximation properties, inverse inequalities, and discrete solution operators.These tools establish discrete smoothing estimates needed for the error bounds.
Semidiscrete scheme: The semidiscrete method approximates the problem in a continuous piecewise linear finite element space on shape-regular, quasi-uniform meshes.The scheme uses the discrete Laplacian, projected source term, and an approximation vh of the initial data.
Homogeneous problem: For smooth initial data v ∈˙H2(Ω), the error is split into a finite element evolution component and a projection component, leading to the first semidiscrete estimate.The initial approximation is chosen as vh = Rhv.
Homogeneous problem: For nonsmooth initial data v ∈˙Hq(Ω), −1 < q ≤1, the analysis replaces the Ritz projection with vh = Phv and introduces a logarithmic factor ℓh = | ln h|.The resulting estimate is stated in Theorem 3.2.
Inhomogeneous problem: For inhomogeneous data f ∈L∞(0, T; ˙Hq(Ω)), −1 < q ≤0, the paper derives semidiscrete error estimates in both L2 and L∞ norms in time.The proof again uses projection splitting and bounds for the source-driven component.

4. A Fully Discrete Scheme

The paper introduces a fully discrete finite-element scheme using finite differences for the time-fractional derivatives, proves unconditional stability, and establishes an error estimate under sufficient solution smoothness.

Discretization: The time interval is uniformly divided with step size τ = T/K to discretize the time-fractional derivatives.The resulting method uses the finite difference discretization introduced in.
Stability: The coefficient sequence satisfies P0 > P1 > ... > 0 and Pj → 0 as j → ∞, supporting the stability argument.The proof uses the monotone decrease of {Pj} and tests the scheme with U n+1.
Discrete scheme: The fully discrete scheme seeks U n+1 ∈ Xh satisfying the discrete fractional operator and Galerkin diffusion equation for every χ ∈ Xh.The right-hand side is evaluated as F n+1 = f(x, t_n+1).
Stability: The fully discrete scheme is unconditionally stable for all n ∈ N.The stability constant c depends only on α and T.
Error analysis: The error estimate is derived assuming that the exact solution u is sufficiently smooth.The analysis compares the continuous and discrete equations, bounds the truncation error, and applies the stability result.
Limitation: For nonsmooth initial data, no error estimate solely in terms of the initial data and right-hand side is known for fully discrete schemes, even in the single-term case.This limitation is stated explicitly in Remark 4.1.

5. Numerical Experiments

The numerical study uses one- and two-dimensional experiments to verify the theoretical error estimates for homogeneous and inhomogeneous problems.

Experimental design: The experiments cover one- and two-dimensional problems.They are used to illustrate the theoretical convergence results.
Experimental design: The numerical tests separately examine homogeneous and inhomogeneous problems.This separation structures the experimental discussion.

5.1. The case of a smooth solution.

For a smooth one-dimensional solution, the experiments isolate temporal discretization error and confirm the theoretically predicted convergence rates for several fractional orders.

Test problem: The smooth test uses v(x) = x(1 − x) and an exact solution u(x, t) = (1 + t^2)(−x^2 + x).The selected source term produces a very smooth exact solution.
Discretization and metric: The computation uses h = 1/N and τ = 1/K, with N chosen large enough that temporal error dominates.Accuracy is measured by the normalized L2 error.
Results: The temporal convergence rates for α = 0.25, 0.5, and 0.95 fully confirm the theoretical result.The rates are reported in Table 1 and plotted in Figure 1.

5.2. Homogeneous problems.

The homogeneous-problem experiments test smooth, nonsmooth, and very weak initial data, confirming predicted spatial rates while revealing deterioration near t = 0 and grid-alignment effects.

Initial data: The nonsmooth and very weak tests use a characteristic-function initial datum and a Dirac delta concentrated at x = 1/2.The characteristic function belongs to H˙^ε for ε ∈ [0, 1/2), while the Dirac datum belongs to H˙^−ε for ε ∈ (1/2, 1].
Experimental design: The spatial experiments make the temporal discretization error negligible by choosing τ = t/(5 × 10^4).Errors are normalized in both L2 and H1 norms.
Spatial convergence: O(h^2) L2 and O(h) H1 convergence are observed for all three α values in the spatial tests.At t = 1, the error decreases as α increases from 0.25 to 0.95.
Nonsmooth data: For nonsmooth data, the numerical results confirm the theoretically predicted rates at t = 1, 0.01, and 0.001.Table 2 reports the errors for α = 0.5 and β = 0.2.
Short-time behavior: With fixed h = 2^−6, the L2 error deteriorates as t → 0 and grows like O(t^−3α/4).This behavior agrees well with the theoretical prediction for v ∈ H˙^(1/2−ε).
Very weak data: For Dirac initial data aligned with the grid, rates are O(h^2) in L2 and O(h) in H1, whereas misalignment gives O(h^3/2) and O(h^1/2).The aligned case is attributed to smoothness on both sides of the support point and favorable finite-element approximation.

5.3. Inhomogeneous problems.

The inhomogeneous-problem experiments test nonsmooth and very weak right-hand sides, confirming the predicted spatial convergence rates while revealing time-dependent error deterioration for nonsmooth data.

Numerical results for example (3b): For the very weak right-hand side involving a Dirac δ1/2(x)-function, grid alignment determines the observed convergence behavior.The aligned-grid results superconverge, whereas the nonaligned-grid results follow reduced theoretical rates.
Numerical results for example (3a): O(h2) and O(h) convergence rates are observed in the L2- and H1-norms, respectively, for the nonsmooth-data example.Errors are reported at t = 1, 0.01, and 0.001.
Numerical results for example (3b): O(h3/2) and O(h1/2) convergence rates occur in the L2- and H1-norms, respectively, when the Dirac singularity is not aligned with the grid.These rates agree with the theoretical prediction.

5.4. Examples in two-dimension.

Two-dimensional unit-square experiments examine nonsmooth and very weak initial data and a nonsmooth right-hand side, with results agreeing with the predicted spatial rates and displaying superconvergence in one case.

Examples in two-dimension: The two-dimensional tests use three examples: nonsmooth initial data, boundary-supported very weak initial data, and a nonsmooth right-hand side.The domain is the unit square and the finite element mesh is generated from equally spaced square subdivisions.
Example (4a): O(h2) and O(h) rates are obtained in the L2- and H1-norms, respectively, for the nonsmooth initial-data example.The result agrees with Theorem 3.2.
Example (4b): Both error norms exhibit super-convergence for the boundary-supported Dirac initial-data example.The corresponding solution profiles and the nonsmooth region are shown at t = 0.1.
Example (4c): The nonsmooth right-hand-side example confirms the theoretical results, with a visibly nonsmooth solution region in its profile.The numerical data are reported in Table 9 and the profile is shown at t = 0.1.

6. Concluding remarks

The paper develops Galerkin finite element schemes for multi-term time-fractional diffusion and establishes analyses covering nonsmooth data, alongside a fully discrete method for smooth solutions.

Concluding remarks: The space semidiscrete Galerkin scheme receives a complete error analysis covering nonsmooth initial data and right-hand sides.The analysis relies on new regularity results for the multi-term equation.
Concluding remarks: A fully discrete scheme uses finite differences for the Caputo fractional derivatives, with stability and error estimates established for smooth solutions.The conclusion specifically identifies the time discretization as a finite difference approximation.
Concluding remarks: One- and two-dimensional numerical experiments confirm the convergence analysis for both smooth and nonsmooth data.The empirical convergence rates agree well with the theoretical predictions.

Appendix A. Proof of Lemma 2.4

The appendix proves a lemma by representing auxiliary functions through inverse Laplace transforms and deforming the Bromwich contour to a Hankel path, while establishing complete monotonicity.

Auxiliary functions: The proof defines an auxiliary function vj(t) and uses Laplace-transform properties to characterize its behavior near t = 0.The initial value is obtained from f(0+) = limz→∞ z f̂(z).
Inverse Laplace representation: The function Ē⃗α(t) is represented by an inverse Laplace integral along the Bromwich path.The transform has a branch point at zero, motivating a cut along the negative real axis.
Analytic structure: The denominator zα + Σk=1^m bk zαk + λj has no zero on the main sheet because the relevant sine terms share the same sign.This zero-free property supports the contour deformation used in the proof.
Contour deformation: The Bromwich path is bent into a Hankel path around the branch cut, and the limit as the inner radius tends to zero yields the representation used in the lemma.The contour starts and ends along the two sides of the negative real axis.
Monotonicity conclusion: The proof concludes that Ē⃗α(t) and vj(t) are completely monotone.The complete-monotonicity conclusion follows from the preceding contour and sign arguments.