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An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. (English) Zbl 1336.65150
The authors first consider an initial boundary problem in a bounded convex polygonal domain $$\Omega\in \mathbb R^d$$, $$d=1,2,3$$, where the time derivative is the left-sided Caputo fractional derivative of order $$\alpha$$, $$0<\alpha<1$$, and the space operator is the Laplacian along with homogeneouos Dirichlet boundary conditions. For the fractional derivative, the L1 scheme of Z.-Z. Sun and X. Wu [Appl. Numer. Math. 56, No. 2, 193–209 (2006; Zbl 1094.65083)] is used for which the local truncation error was shown to be $$O(\tau^{2-\alpha})$$ in the case that the solution is twice continuously differentiable. The authors first show by a numerical experiment that this is not the order of convergence of the scheme, neither for non-smooth data nor for smooth ones. They then show by using linear finite elements in space and the resolvent of the semi-discretized problem that the $$L_2$$-error for $$L_2$$ initial data is of order $$O(h^2t^{-\alpha})$$. For the fully discretized problem on an equidistant time grid with step $$\tau$$, applying the L1 scheme, they further show that the error for $$L_2$$ initial data (compared to the semi-discretized case) is of the order $$O(\tau t^{-1})$$ which improves to $$O(\tau t^{\alpha-1})$$ for smooth data.
In short, for fixed time and $$h=O(\tau^{1/2})$$, one has first-order convergence in both cases of smooth and non-smooth data. This is confirmed by a series of numerical experiments. Finally, an error estimate is obtained for an equation with the Laplacian replaced by a more general operator like the Riemann-Liouville fractional derivative of order $$\beta\in(\frac32,2)$$. This estimate is also confirmed by numerical experiments.

##### MSC:
 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35R11 Fractional partial differential equations
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