Error Analysis of a Finite Difference Method on Graded Meshes for a Time-fractional Diffusion Equation
Update: 2018-02-24 09:23:43      Author: yangjuan@csrc.ac.cn

The numerical solution of initial-boundary value problems with a fractional time derivative of order a, where 0 < a <1, has received much attention in the last few years. If a = 1, then one has a standard parabolic partial differential equation, while 0 < a <1 means that the equation models an anomalous diffusion process (subdiffusion). Given smooth data, typical solutions of this problem are smooth in the spatial variable, and in the time variable for t > 0, but at the initial time t=0 the solution has a weak singularity: it is continuous but its integer-order time derivatives blow up at t approaches zero. 

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The figure shows a typical solution, and a cross-section of this solution along a line x=constant; clearly there is a singularity at t = 0. But the vast majority of published analyses of numerical methods for this class of problem assume that the solution is a smooth function of both variables on all of the domain, i.e., they ignore the effect of the initial singularity. This global smoothness assumption limits severely the class of problems to which their theory can be applied; see [1]. 

The paper [2] is the first to give a finite difference analysis of a numerical method for these problems that is valid for problems that have a weak singularity at t = 0. In [2] one uses a uniform mesh in space and a special graded mesh in time (this mesh grading is often used in the solution of Volterra weakly integral equations, which are closely related to the fractional-derivative problem). A new discrete stability inequality is derived, showing how the solution at each time level depends on the data at earlier times; this is combined with a consistency error analysis to obtain a final convergence result in the discrete maximum norm that is sharp (as demonstrated by numerical experiments) and shows exactly how the error depends on the degree of mesh grading used. Consequently one can prescribe a priori an optimal mesh grading for the problem. 

Reference [2] has attracted a great deal of attention in the fractional-derivative community: a copy of it on Researchgate has had more than 700 Reads.

References:

[1] M.Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal. 19 (2016), no.6, 1554–1562.  DOI: https://doi.org/10.1515/fca-2016-0080

[2] M.Stynes, E.O'Riordan & J.L.Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), 1057–1079.

DOI: 10.1137/16M1082329



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