Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions

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 framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the L∞ and L 2 norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.1991 Mathematics Subject Classification. Primary 65M06, 65M15, 65M60.

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

“…This implies that the optimal grading parameter for global accuracy is r = (2 − α)/α. Note that similar global error bounds were obtained in [10,8,14].…”

“…By contrast, [10, Theorem 3.1] (obtained by means of a discrete Grönwall inequality) gives a somewhat similar, but less sharp error bound for graded meshes, as (in our notation) it involves the term O(τ α ) = O(M −αr ), so, e.g., [10, (3.17)] requires (in our notation) r = (2 − α)/α to attain the optimal convergence rate in positive time. Note that for r = 1, we have E m M −1 , so our error bound is consistent with [4,6,8] and is sharper than [10, (3.17…”

“…In particular, the latter implies that the errors are M − min{r,2−α} for r = 2 − α and M −(2−α) ln M for r = 2 − α. The maximum errors and corresponding convergence rates for various α and r are given in [14,8], and they confirm the conclusions of Remark 3.3, which predicts from Fractional-order parabolic test problem: L 2 (Ω) errors at t = 1 (odd rows) and computational rates q in M −q (even rows) for the L1 method with r = (2 − α)/.9 and the Alikhanov method with r = 2, spatial DOF=398410 L1 method, r = 2−α the pointwise bound (3.2) that the global errors are M − min{αr,2−α} . Furthermore, in Fig.…”

“…The generality of our approach is demonstrated in the second part of the paper by extending the stability and error analysis to higher-order Alikhanov-type schemes [1]. Similarly to [14,8,2], our main interest will be in graded temporal meshes as they offer an efficient way of computing reliable numerical approximations of solutions singular at t = 0, which is typical for (1.3). It should be noted that these three papers are concerned with global-in-time error bounds on graded meshes.…”

High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation

“…with h ≤ h 1 = (C Ω C * 2 ) −1 . Using (4.3) and Theorem 3.1, we get [33,34]). Taking = |U n h − u n |, 1 = , 2 = U n − u n in (1.4), we get…”