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Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations
Kassem Mustapha, William McLean
TL;DR
The paper asks whether piecewise-linear DG time discretization can achieve superconvergence for fractional diffusion and wave equations despite solution singularities near t=0. It combines DG time stepping with conforming finite elements in space and proves nodal and postprocessed superconvergence on graded meshes. Theoretical and numerical results identify the achievable rates and their dependence on α.
Problem
Fractional diffusion and wave equations can have solution derivatives singular near t=0, limiting the reliability of formally high-order time discretizations.
Method
The paper analyzes a piecewise-linear discontinuous Galerkin method in time with graded time meshes and continuous piecewise-linear finite elements in space.
Results
O(k^{3+2α−}) nodal and uniformly-in-time postprocessed error bounds are proved, while experiments observe O(k^{3+α−}+h^2) accuracy when logarithmic factors are ignored.
Takeaways & Limitations
Graded meshes enable high temporal convergence despite initial singularities, and simple Lagrange interpolation extends superconvergence from time nodes to all t.
Takeaways & Limitations
The theoretical analysis assumes spatial-operator conditions including a strictly positive-definite operator, and the tested exact solution fails one of the stated regularity assumptions.
Abstract
from arXiv · showhide
We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-α}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<α<0$) or wave ($0<α<1$) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+α_-}+h^2\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $α_-=\min(α,0)\le0$ and $\ell(k)=\max(1,|\log k|)$. Here, we generalize a known result for the classical heat equation (i.e., the case $α=0$) by showing that at each time level $t_n$ the solution is superconvergent with respect to $k$: the error is of order $(k^{3+2α_-}+h^2)\ell(k)$. Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation for any $t$. Numerical experiments indicate that our theoretical error bound is pessimistic if $α<0$. Ignoring logarithmic factors, we observe that the error in the DG solution at $t=t_n$, and after postprocessing at all $t$, is of order $k^{3+α_-}+h^2$.
1. Introduction.
The paper studies piecewise-linear DG time discretization for fractional diffusion and wave equations, targeting superconvergence despite solution singularities near t=0. It proves nodal and postprocessed superconvergence using appropriately graded time meshes and extends the classical heat-equation result.
- Motivation: Fractional diffusion models use −1 < α < 0, while fractional wave models use 0 < α < 1.These problems arise in applications including anomalous sub-diffusion and viscoelastic wave propagation.
- Motivation: Higher-order derivatives of the exact solution are typically singular as t → 0, so formally high-order methods may converge slowly.Non-uniform time steps are used to accommodate this low regularity.
- Results: O(k^{3+2α−}) nodal error is proved for graded time meshes, extending known superconvergence results for the classical heat equation.The paper restricts attention to the piecewise-linear DG method, q = 2.
- Results: O(k^{3+2α−}) accuracy is obtained for a postprocessed solution uniformly in t through simple interpolation.The postprocessed result applies at all times rather than only at nodal values.
2. Preliminaries.
The preliminaries define the spatial operator, graded time mesh, piecewise-linear DG trial space, fractional time operator, and discontinuous quasi-interpolant. They also establish well-posedness, unconditional stability, and interpolation error estimates used later.
- Spatial operator: The spatial operator A is assumed self-adjoint, strictly positive-definite, and equipped with a complete ordered eigensystem.These assumptions include the Dirichlet Laplacian on suitable bounded domains.
- Time discretization: The time mesh 0 = t0 < t1 < ··· < tN = T uses steps kn and maximum step k.The DG trial space consists of functions that are piecewise linear on each time interval.
- Fractional operator: For −1 < α < 1, the fractional DG procedure determines U using the nonlocal operator Bα and test functions in P1.Each time step involves all previous time levels, although the resulting sum admits a fast algorithm.
- Stability: The DG method is unconditionally stable and has a unique solution U satisfying the discrete equations.The solution also belongs to D(A) for t > 0.
- Quasi-interpolant: The discontinuous quasi-interpolant Π− is defined through interval moments and endpoint conditions, with integral representations for its interpolation error.These estimates support the later duality-based error analysis.
3. Dual problem.
The dual problem provides a representation of the nodal error and supplies regularity estimates for the adjoint solution. Combining dual stability with DG approximation estimates yields the error bound needed for nodal superconvergence.
- Dual formulation: The dual problem reverses the terminal-value perspective and uses the adjoint fractional operator B∗α.For negative α, the adjoint operator has a weak singularity near the time nodes.
- Nodal error representation: A representation theorem expresses the nodal error through the primal and dual solutions and their DG approximations.This representation is the basis for the nodal-error analysis.
- Dual regularity: The dual solution satisfies regularity estimates that control its time derivatives in terms of the terminal value zT.Time reversal converts the dual problem into a forward fractional evolution problem.
- Dual error analysis: The dual DG error is analyzed by splitting A−1(Z − z) into projection and discrete-solution components.The components are ζ = A−1(Π+z − z) and Θ = A−1(Z − Π+z).
- Dual error analysis: The resulting dual estimates combine stability, interpolation, and adjoint regularity to support the final nodal error bound.The analysis applies for −1 < α < 1.
4. Nodal superconvergence.
The piecewise-linear DG method achieves nodal superconvergence on suitably graded non-uniform time meshes. Under the stated assumptions, the nodal error has order k^{3+2α_-}, up to time-dependent logarithmic and solution-regularity factors.
- Nodal superconvergence: The required time meshes are non-uniform and satisfy assumptions expressed through a grading parameter γ and threshold γ∗.The threshold is γ∗ = (2 + α_-)/σ.
- Nodal superconvergence: The nodal error is O(k^{3+2α_-}) when the mesh grading satisfies γ > γ∗ = (2 + α_-)/σ.The bound includes the factor t_n^{2α}ℓ(t_n/k_n).
5. Postprocessing.
A simple Lagrange-interpolation postprocessing of the DG solution produces a superconvergent approximation uniformly for all times. With sufficiently strong mesh grading, its time error is of order k^3, subject to the stated regularity and mesh conditions.
- Postprocessing: Lagrange interpolation postprocesses the discontinuous DG solution into a globally superconvergent approximation U♯.The construction uses linear interpolation on the first two subintervals and backward quadratic interpolation thereafter.
- Postprocessing: The interpolation analysis requires a regularity assumption involving a positive exponent σ♯ and a local quasi-uniformity condition on subsequent subintervals.The mesh condition controls the stability and interpolation error of the postprocessing operator.
- Postprocessing: The postprocessed solution is superconvergent uniformly in t when the mesh satisfies the stated conditions.The proof decomposes the error into interpolation error and the interpolated DG error.
- Postprocessing: The postprocessed error is O(k^3) when γ ≥ 3/σ♯; for weaker grading, the bound is O(k^{γσ♯}).The rate is stated piecewise according to the grading parameter.
6. Spatial discretization.
The fully discrete method combines the time DG scheme with continuous piecewise-linear finite elements on a quasiuniform spatial mesh. Its nodal error retains the temporal superconvergence rate while adding a spatial term of order h^2.
- Spatial discretization: The spatial analysis assumes A = −∇^2 on a bounded convex or C2 domain with homogeneous Dirichlet boundary conditions.These assumptions provide the H2 regularity used for the finite element error estimate.
- Spatial discretization: The spatial discretization uses continuous piecewise-linear finite elements on a quasiuniform mesh with maximum element diameter h.The discrete space consists of continuous piecewise-linear functions, and the fully discrete solution is continuous in space but discontinuous in time.
- Spatial discretization: The fully discrete DG method is defined by restricting the time-DG formulation to the finite element space S_h.The initial value is chosen using the L2-projector P_h, while the Ritz projector R_h supplies the spatial approximation framework.
- Fully-discrete nodal error: The fully discrete nodal error has the same temporal bound as the semidiscrete method with an additional spatial contribution of order h^2.The dual-problem analysis establishes the spatial approximation component used in the final estimate.
- Fully-discrete nodal error: The fully discrete nodal result requires the same graded-time-mesh condition γ > γ∗ = (2 + α_-)/σ used for temporal nodal superconvergence.The theorem applies under the stated regularity assumptions on the solution.
7. Numerical results.
Numerical tests use a one-dimensional model problem to examine graded time meshes, nodal errors, and postprocessed global errors for fractional diffusion and wave equations. They find sharper-than-theoretical convergence for α<0, while non-uniform meshes improve accuracy except near the endpoint values of α.
- Experimental setup: The tests use a one-dimensional problem with homogeneous Dirichlet conditions, a graded time mesh, and a uniform spatial mesh chosen so h^2≈k^3.The choice M=⌈N^3/2⌉ makes the time-discretization error dominant.
- Exact solution and regularity: The exact solution satisfies the stated regularity condition with σ=(1+ε)(1+α)<5 but fails the second theoretical regularity assumption.This makes the experiments a test beyond the full scope of the analysis.
- Nodal errors: O(k^(3+α_-)) is observed for α<0, exceeding the theoretical O(k^(3+2α_-)) nodal bound.For α=−0.3, the highest observed rate is O(k^(3+α_-)); the right-hand nodal value is at best O(k^2).
- Mesh grading: The maximum nodal error is minimized near γ≈(3+α_-)/σ, while non-uniform time steps generally help except when α is close to −1 or 1.For α=0.2, the reported best grading is approximately γ=2.
- Global error after postprocessing: After Lagrange-interpolation postprocessing, the global error follows the predicted O(k^(3+α_-)) behavior in the numerical tests.Tables 7.5 and 7.6 report the uniform postprocessed DG error for α=−0.3 and α=+0.3.
8. Concluding remarks.
The piecewise-linear DG method achieves nodal superconvergence for fractional diffusion and wave equations, with postprocessing extending high accuracy to all times despite solution singularities near t=0.
- O(k^{3+α_-}) convergence is achieved with appropriately graded time steps despite singular derivatives of the exact solution near t=0.The observed rate applies to generic regular data u0 and f.
- Superconvergence is proved at the time nodes, generalizing a known result for the classical heat equation.
- Postprocessing extends the same high accuracy from the nodes to every t.
- The spatial discretization with continuous piecewise-linear finite elements contributes essentially O(h^2) additional error.
- Numerical experiments indicate that the theoretical error bounds are sharp for α > 0 but pessimistic for α < 0.