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A discontinuous Petrov-Galerkin method for time-fractional diffusion equations
Kassem Mustapha, Basheer Abdallah, Khaled Furati
TL;DR
High-order time discretizations for time-fractional subdiffusion face singular solution behavior near t=0. The paper analyzes a time-stepping DPG method with continuous finite elements in space; its theory gives a global error order, while numerical experiments indicate optimal convergence in both variables.
Problem
High-order time methods can lose fast convergence because the solution typically has singular behavior near t=0.
Source: §1.1 · p. 2Method
The paper analyzes a time-stepping discontinuous Petrov-Galerkin method combined with continuous finite elements in space for time-fractional diffusion.
Source: §1 · p. 1Results
The analysis gives O(k^(m+α/2) + h^(r+1)); numerical results indicate optimal convergence in both time and space.
Source: §1.1 · p. 2, §4.6 · p. 13Takeaways & Limitations
For the studied method, numerical experiments show better time convergence than the theoretical estimate, indicating that the error bound is pessimistic.
Source: §1.1 · p. 2, §4.6 · p. 13The analysis assumes sufficient solution regularity, including u∈W^{1,1}((0,T);H^2(Ω)); further regularity assumptions are imposed later.
Source: §1 · p. 1
Abstract
from arXiv · showhide
We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness and stability of approximate solutions, and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near~$t=0$ to compensate the singular (temporal) behaviour of the exact solution near $t=0$ caused by the weakly singular kernel, but the spatial mesh is quasiuniform. In the $L_\infty((0,T);L_2(\Omega))$-norm ($(0,T)$ is the time domain and $\Omega$ is the spatial domain), for sufficiently graded time meshes, a global convergence of order $k^{m+\alpha/2}+h^{r+1}$ is shown, where $0<\alpha<1$ is the fractional exponent, $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, and $m$ and $r$ are the degrees of approximate solutions in time and spatial variables, respectively. Numerical experiments indicate that our theoretical error bound is pessimistic. We observe that the error is of order ~$k^{m+1}+h^{r+1}$, that is, optimal in both variables.
1. Introduction.
The paper develops a time-stepping discontinuous Petrov-Galerkin method coupled with continuous finite elements for time-fractional diffusion. It addresses singular behavior near t=0 with nonuniform time steps and analyzes the method’s stability and convergence.
1. Introduction.
- The DPG-FE method combines time-stepping discontinuous Petrov-Galerkin discretization with continuous finite elements in space for time-fractional diffusion.The model describes slow or anomalous subdiffusion, including settings where particle flux depends on the history of the density gradient.Source: §1 · p. 1
1.1. Motivation and outline of the paper.
- High-order time methods can lose fast convergence because the solution is typically singular near t=0; the proposed DPG method uses nonuniform time steps to address this behavior.The scheme is analyzed for existence, uniqueness, and stability, with a stated error order of O(k^(m+α/2) + h^(r+1)) in L∞((0,T);L2(Ω)).Source: §1.1 · p. 2
1.2. Literature review.
- The review surveys finite-difference, spectral, ADI, compact-difference, and FEM approaches to related formulations [38],,,,,,,,, [11].The cited works report a range of convergence and stability analyses under different solution regularity assumptions.Source: §1.2 · p. 3
2. Numerical scheme.
The fully discrete DPG-FE scheme uses a possibly nonuniform time partition and a quasiuniform spatial finite-element mesh. Its time-dependent trial and test spaces use polynomial degrees m and m−1, respectively.
2. Numerical scheme.
- The scheme partitions [0,T] into possibly nonuniform intervals and uses continuous, piecewise polynomials of degree at most r on a quasiuniform conforming spatial mesh.The spatial finite elements are triangular, with maximum diameter h.Source: §2 · pp. 3–4
- The trial space consists of continuous-in-time functions that are degree-m polynomials on each time interval, while the test space uses degree m−1.The piecewise time derivative of a trial-space function belongs to the test space.Source: §2 · p. 4
- The discrete solution is initialized with the Ritz projection of u0 onto the spatial finite-element space.The scheme seeks Uh in the trial space with Uh(0)=Rhu0.Source: §2 · p. 4
3. Well-posedness of the DPG-FE scheme.
The paper establishes existence, uniqueness, and stability for the discrete DPG-FE solution under stated data assumptions. The analysis uses positivity, coercivity, and continuity properties of the fractional derivative operator.
3. Well-posedness of the DPG-FE scheme.
- The well-posedness analysis relies on positivity, coercivity, and continuity properties of the fractional derivative operator.The uniqueness argument uses coercivity to conclude that the difference between two candidate solutions vanishes on the first time interval.Source: §3 · pp. 4–5
- The discrete solution exists and is unique when f∈L2((0,T);L2(Ω)) and u0∈H1_0(Ω).The proof uses the finite-dimensional problem on each time subdomain and establishes uniqueness successively in time.Source: §3 · pp. 4–5
- The stability theorem assumes f∈H1((0,T);L2(Ω)) and u0∈H1_0(Ω), and establishes stability for the DPG-FE solution at each time level.The supplied theorem passage does not state the bound’s explicit form.Source: §3 · p. 5
4. Error Analysis.
The error analysis combines projection estimates and an error decomposition with time meshes graded near t=0 to address solution singularity. The theoretical bounds are suboptimal in time but optimal in space, while numerical results indicate optimal convergence in both.
4.1. Projection and errors.
- The projection Π matches u at interval endpoints and satisfies moment orthogonality; on the first interval it is linear to accommodate the singularity at t=0.The moment condition uses polynomials of degree at most m−2.Source: §4.1 · p. 6
4.2. Error decomposition and an interesting bound.
- The error is decomposed into ζ, Πξ, and η; under u∈W^1,1((0,T);H^2(Ω)), the main task is bounding ζ.The decomposition uses the DPG-FE solution, Ritz projection, and time projection.Source: §4.2 · p. 7
4.3. Regularity and time meshes.
- Because u is singular near t=0, uniform steps may not deliver O(k^(m+1)) time accuracy; the analysis uses nonuniform meshes concentrated near zero.The paper notes that high-order methods on uniform meshes typically should not be expected to exceed O(k^σ) globally in the L∞(0,T)-norm.Source: §4.3 · p. 9
- For γ≥(2m+1+α)/(α+2σ−1), one fractional-error term is bounded by Ck^(2m+α); for γ≥(m+1)/(δ−1), another is bounded by Ck^(2m+1).Source: §4.4 · p. 9, §4.5 · p. 11
4.6. The error estimates.
- Theoretical time convergence is short by order 1−α/2 of optimal, while spatial convergence is optimal under sufficient solution regularity; numerical results indicate optimal rates in both variables.The estimates concern the DPG-FE solution.Source: §4.6 · p. 13
5. Numerical results.
The experiments test the DPG-FE scheme’s time and space convergence on two exact solutions with different temporal smoothness. Both examples show optimistic time-convergence rates on graded meshes, while Example 1 also tests spatial convergence.
5. Numerical results.
- The experiments approximate the L∞((0,T);L2(Ω)) error by sampling on a finer time grid and applying composite Gauss quadrature in space.For large q, the sampled maximum over Gq approximates the target norm.Source: §5 · p. 14
5.1. Example 1.
- The examples use exact solutions t^(α+1)sin(πx) and t^(1−α)sin(πx), with the latter described as less smooth in time.The corresponding regularity parameters are σ=α+1, δ=α+2 for Example 1 and σ=1−α, δ=2−α for Example 2.Source: §5.1 · p. 14, §5.2 · p. 14
- For Example 1, the global error is bounded by Ck^min{γ(α+1),m+1} for γ≥1 and reaches the optimal rate when γ≥(m+1)/(α+1).The reported rate is k^(α+1)γ for 1≤γ≤(m+1)/(α+1); errors are also plotted against N for different γ and m.Source: §5.1 · p. 14, Table 5.1 · p. 15, Figure 5.1 · p. 15
- For Example 1, spatial finite elements of degrees r=1, 2, and 3 are tested with time errors suppressed, and the reported spatial order is h^(r+1).The experiment uses a uniform spatial mesh.Source: §5.1 · p. 14, Table 5.2 · p. 16
5.2. Example 2.
- For Example 2, the global error is bounded by Ck^min{γ(1−α),m+1} for γ≥1 and is optimal for γ≥(m+1)/(1−α).The observed optimality threshold is relaxed relative to the theoretical requirement γ≥2m+α+1.Source: §5.2 · pp. 15–16