Abstract
For time-fractional parabolic equations with a Caputo time derivative of order α∈(0,1), we give pointwise-in-time a posteriori error bounds in the spatial L2 and L∞ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates 2−α in the presence of solution singularities.
Highlights
Consider a fractional-order parabolic equation, of order α ∈ (0, 1), of the formDtαu + Lu = f (x, t) for (x, t) ∈ Ω × (0, T ], (1.1)subject to the initial condition u(·, 0) = u0 in Ω, and the boundary condition u = 0 on ∂Ω for t > 0
Subject to the initial condition u(·, 0) = u0 in Ω, and the boundary condition u = 0 on ∂Ω for t > 0. This problem is posed in a bounded Lipschitz domain Ω ⊂ Rd, and involves a spatial linear second-order elliptic operator L
The main results of the paper are as follows
Summary
Consider a fractional-order parabolic equation, of order α ∈ (0, 1), of the form. subject to the initial condition u(·, 0) = u0 in Ω , and the boundary condition u = 0 on ∂Ω for t > 0. It suffices to check that DtαE1(t) = {Γ (1 − α)}−1 t−1ρ(τ /t), as (Dtα + λ)E1(t) = R1(t) ≥ ∥Rh∥, so an application of Corollary 2.3 immediately yields the desired bound ∥e(·, t)∥ ≤ E1(t) ≤ tα−1. Theorem 2.2 and its two corollaries immediately apply to uh once it is reset to u0 at t = 0 (after which, it is worth noting, uh becomes right-discontinuous at t = 0) This modification of uh needs to be reflected in the computation of the residual Rh as follows. An inspection of the proof shows that this condition is only required to apply [10, Theorem 1] (the maximum principle for Dtα) The proof of the latter relies on the representation of type (2.4) and remains valid under our weaker assumptions.
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