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Error estimates for a semidiscrete finite element method for fractional order parabolic equations
Bangti Jin, Raytcho Lazarov, Zhi Zhou
TL;DR
The paper addresses the lack of finite element error estimates that are optimal for limited regularity in time-fractional diffusion. It analyzes semidiscrete Galerkin and lumped-mass Galerkin methods and establishes regularity-aware results, including nonsmooth initial data, with further improvements for special meshes and postprocessed gradients.
Problem
Existing fractional-diffusion numerical analyses largely assume smooth solutions, leaving optimal error estimates for nonsmooth initial data insufficiently addressed.
Method
The paper analyzes semidiscrete Galerkin and lumped-mass Galerkin finite element methods with piecewise linear functions on quasi-uniform meshes and related special triangulations.
Results
The methods achieve optimal regularity-dependent estimates for smooth and nonsmooth data, while special meshes support almost optimal lumped-mass estimates and gradient superconvergence after postprocessing.
Takeaways & Limitations
The analysis extends finite element error estimates for fractional diffusion to limited initial-data regularity and identifies mesh and postprocessing settings that improve convergence behavior.
Takeaways & Limitations
The theory assumes quasi-uniform meshes and does not cover weak initial data such as the Dirac delta function.
Abstract
from arXiv · showhide
We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation $\partial^α_t u - \De u =0$ ($0< α< 1$) with initial condition $u(x,0)=v(x)$ and a homogeneous Dirichlet boundary condition in a bounded polygonal domain $Ω$. We shall study two semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin FEM, by using piecewise linear functions. We establish optimal with respect to the regularity of the solution error estimates, including the case of nonsmooth initial data, i.e., $v \in L_2(Ω)$.
1. Introduction
The paper develops semidiscrete finite element error analysis for homogeneous time-fractional diffusion, targeting estimates that remain optimal for limited initial-data regularity. It studies standard and lumped-mass Galerkin methods, including nonsmooth data and special meshes.
- Problem setting: The paper analyzes the fractional diffusion problem with 0 < α < 1 on bounded polygonal domains, where the Caputo derivative models anomalous subdiffusion.As α approaches one, the fractional derivative recovers the standard first-order derivative.
- Motivation: Earlier numerical analyses generally assumed sufficiently smooth solutions and did not establish error estimates optimal with respect to data regularity.This gap excludes solutions with limited regularity, including cases arising from nonsmooth initial data in inverse problems.
- Methods: The goal is optimal-regularity error analysis for semidiscrete Galerkin and lumped-mass Galerkin FEMs on convex polygonal domains using piecewise linear finite element spaces.The meshes are assumed quasi-uniform, and the initial-data cases include v ∈ ˙Hq(Ω) for q = 0, 1, 2.
- Galerkin FEM: For smooth initial data, the standard Galerkin method attains the same error bound as the standard parabolic problem uniformly for t ≥ 0.The result is stated as a main theorem of the paper.
- Galerkin FEM: For nonsmooth v ∈ L2(Ω), the standard Galerkin estimate deteriorates as t approaches zero and includes an additional logarithmic factor ℓh = |ln h|.This parallels the standard parabolic nonsmooth-data estimate but is derived for quasi-uniform meshes.
- Lumped-mass FEM: The lumped-mass method has the smooth-data rate, almost optimal gradient estimates for v ∈ ˙H1(Ω) and v ∈ L2(Ω), and special-mesh improvements for nonsmooth data.On general quasi-uniform meshes, its nonsmooth-data L2 error is suboptimal of order O(hℓht^-α), while special symmetric meshes yield an almost optimal estimate.
- Superconvergence: For smooth initial data on planar domains with special meshes, postprocessing produces superconvergence for the gradient error in both Galerkin and lumped-mass approximations.The result is established in Theorem 5.1.
2. Preliminaries
The preliminaries establish the Mittag-Leffler solution representation, fractional smoothing estimates, and finite-element projection properties used in the error analysis. They also identify the fractional equation’s weaker smoothing compared with the standard parabolic problem.
- Mittag-Leffler function: The Mittag-Leffler functions Eα,1(-λt^α) and t^α−1Eα,α(-λt^α) appear in the initial-value and nonhomogeneous solution operators.The exponential function is recovered as E1,1(z) = e^z.
- Function spaces: The spaces ˙Hq(Ω) are defined using the Dirichlet Laplacian eigenpairs, with ˙H0(Ω) corresponding to L2(Ω) and ˙H−q defined by duality.The eigenfunctions form an orthonormal basis in L2(Ω).
- Solution representation: The homogeneous solution is represented by u(t) = E(t)v, while E(t) and ¯E(t) represent the homogeneous and nonhomogeneous parts of the general solution.The operator E(t) is defined through the Dirichlet eigenpairs of the Laplacian.
- Smoothing estimates: The solution satisfies stability and smoothing estimates that connect spatial regularity at time t to the regularity of the initial data.These estimates are the key analytic input for the finite element error analysis.
- Smoothing estimates: For the fractional problem, the smoothing estimate with ℓ = 1 requires p ≤ q, unlike the corresponding standard parabolic estimate.This restriction reflects the fractional equation’s limited smoothing and influences the finite element error bounds.
- Finite element projections: The orthogonal L2 projection Ph and Ritz projection Rh satisfy O(hq) approximation bounds in combined L2 and H1-seminorm error for q = 1, 2.For globally uniform meshes, Ph is also stable in ˙H1.
3. Semidiscrete Galerkin FEM
The semidiscrete Galerkin analysis uses discrete solution-operator smoothing to derive error estimates for smooth and nonsmooth initial data, including v ∈ L2(Ω).
- Discrete formulation: The method analyzes the spatially discrete problem through discrete solution operators and their smoothing properties.These properties are discrete analogues of the continuous smoothing estimates and underpin the convergence analysis.
- Smoothing estimates: The discrete smoothing estimate permits flexible regularity indices, which is essential for low-regularity initial data.For p = 0, 1, q may range over p−2 ≤ q ≤ p.
- Smooth initial data: For smooth initial data, the Galerkin FEM achieves the same error bound as the standard parabolic problem uniformly for t ≥ 0.The result is stated for initial data in the relevant smoothness class and uses the Ritz projection.
- Nonsmooth initial data: For nonsmooth initial data v ∈ L2(Ω), the analysis uses the L2-projection and obtains an almost optimal estimate with ℓ_h = |ln h|.The error is split into projection and discrete-evolution components, with the logarithmic factor arising from the estimate.
- General elliptic operators: The framework extends to fractional-order problems with more general coercive elliptic operators under stated regularity assumptions.The extension includes spatially varying coefficients and more general boundary conditions.
4. Lumped mass finite element method
The lumped mass Galerkin FEM is formulated using an approximate inner product and a discrete Laplacian, then analyzed through quadrature-error operators and smoothing estimates.
- Discrete formulation: The lumped mass method replaces the standard L2 inner product with a quadrature-based inner product and defines a corresponding discrete Laplacian.Its operator form is t^α ū_h(t) − Δ̄_h ū_h(t) = f̄_h(t), with projected initial data.
- Quadrature error: The quadrature error operator Q_h represents the mass-lumping error and supports the subsequent error analysis.It is defined through the difference between the approximate and exact inner products.
- Smooth initial data: For smooth initial data, the lumped mass method has the same convergence rate as the standard Galerkin FEM.The theorem applies when the initial finite element value is chosen using the Ritz projection.
- Nonsmooth initial data: On general quasi-uniform meshes, the L2 error for v ∈ L2(Ω) is only bounded suboptimally by O(hℓ_h t^−α).The estimate applies to nonsmooth data and reflects deterioration as t approaches zero.
- Nonsmooth initial data: For v ∈ Ḣ1(Ω) and v ∈ L2(Ω), the method provides an almost optimal gradient error estimate.An additional logarithmic factor ℓ_h appears in the nonsmooth-data estimates.
- Mesh conditions: Symmetric meshes satisfy the additional quadrature condition needed for optimal L2 convergence with nonsmooth data.Without that condition, only a suboptimal O(h) L2 convergence rate was established.
5. Special meshes
Special mesh structure enables superconvergent gradient recovery and higher-order estimates for the semidiscrete Galerkin method, with analogous implications for lumped mass FEM.
- Superconvergence: The superconvergence property is available for special meshes and sufficiently smooth solutions in H3(Ω).Examples include triangulations in which every two adjacent triangles form a parallelogram.
- Mesh structure: Strongly uniform triangulations, where adjacent triangles form parallelograms, provide the mesh structure used for superconvergent gradient recovery.The recovery operator is applied to obtain a higher-order approximation of the gradient.
- Error estimate: The resulting higher convergence-rate estimate applies to the semidiscrete Galerkin method for smooth initial data.The proof combines gradient recovery properties with estimates for the projection error.
- Lumped mass consequence: Strongly regular triangulations are symmetric at internal vertices and therefore yield optimal L2 convergence for nonsmooth data.This conclusion is linked to the symmetric-mesh result for the lumped mass method.
6. Numerical results
The numerical experiments test Galerkin and lumped mass finite element approximations across smooth, intermediate, nonsmooth, and very weak initial data. Observed convergence rates generally agree with the theoretical estimates, while the Dirac-delta case remains outside the theory.
- Experimental setup: The experiments use one-dimensional test problems with five data classes, including smooth, intermediate, nonsmooth, variable-coefficient, and Dirac-delta initial data.The exact solutions for several examples involve Mittag-Leffler functions, and piecewise linear finite element spaces are used.
- Experimental setup: The reference solution uses a weighted finite-difference approximation in time with τ = 1.0 × 10^-6, making temporal discretization error negligible for comparisons.The local truncation error is bounded by Cτ^(2−α) for sufficiently smooth solutions and constant time steps.
- Smooth initial data: Smooth-data experiments produce slopes of 2 and 1 for the L2- and H1-norm errors, respectively, across α = 0.1, 0.5, and 0.95 at t = 1.The recovered gradient additionally exhibits O(h^2) convergence in one dimension.
- Intermediate smoothness: For intermediate regularity, the L2- and H1-norm error curves again have slopes 2 and 1 at α = 0.5 and t = 1.The data lie in H^1_0(Ω) ∩ H^(3/2−ε)(Ω), representing an intermediate smoothness case.
- Method comparison and very weak data: Galerkin and lumped mass methods yield almost identical results for the characteristic-function example, so the numerical presentation focuses on the lumped mass method.For Dirac-delta data, which the theory does not cover, the H1 error nevertheless converges as O(h^(3/2)) and good L2- and H1-norm rates are observed.