Error Analysis for a Fractional-Derivative Parabolic Problem on Quasi-Graded Meshes using Barrier Functions

Abstract

An initial-boundary value problem with a Caputo time derivative of fractional order α ∈ (0, 1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semi-discretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.It should be noted that while the explicit inverse of D α t is easily available, the proof of (1.2) for any discrete operator is quite non-trivial. As an alternative, discrete Grönwall inequalities were recently employed in the error analysis of L1-and Alikhanov-type schemes [10,11,12]. However, this approach involves intricate evaluations and, furthermore, yields less sharp error bounds (see Remarks 3.2 and 4.11 for a more detailed discussion). Our approach is entirely different and is substantially more concise as we obtain (1.2) using clever barrier functions (building on the ideas from [8, Appendix A]), while the numerical results indicate that our error bounds are sharp in the poitwise-in-time sense.The following fractional-order parabolic problem is considered:This problem is posed in a bounded Lipschitz domain Ω ⊂ R d (where d ∈ {1, 2, 3}). The spatial operator L here is a linear second-order elliptic operator:

Pointwise-in-time a posteriori error control for time-fractional parabolic equations

“…1 & 2. For R 0 , the errors on the adaptive meshes were compared with the errors on the optimal graded meshes {t j = T (j/M ) r } M j=0 with r = (2 − α)/α [3,5,9] for the same values of M . We observe that in both cases the optimal global rates of convergence 2−α are attained.…”

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Global dynamics for a class of reaction–diffusion multigroup SIR epidemic models with time fractional-order derivatives

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On stability and convergence of L2-1$_σ$ method on general nonuniform meshes for subdiffusion equation

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