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Finite difference method as a popular numerical method has been widely usedto solve fractional diffusion equations. In the general spatial error analyses,an assumption $u\in C^{4}(\bar{\Omega})$ is needed to preserve$\mathcal{O}(h^{2})$ convergence when using central finite difference scheme tosolve fractional sub-diffusion equation with Laplace operator, but thisassumption is somewhat strong, where $u$ is the exact solution and $h$ is themesh size. In this paper, a novel analysis technique is proposed to show thatthe spatial convergence rate can reach$\mathcal{O}(h^{\min(\sigma+\frac{1}{2}-\epsilon,2)})$ in both $l^{2}$-norm and$l^{\infty}$-norm in one-dimensional domain when the initial value and sourceterm are both in $\hat{H}^{\sigma}(\Omega)$ but without any regularityassumption on the exact solution, where $\sigma\geq 0$ and $\epsilon>0$ beingarbitrarily small. After making slight modifications on the scheme, acting onthe initial value and source term, the spatial convergence rate can be improvedto $\mathcal{O}(h^{2})$ in $l^{2}$-norm and$\mathcal{O}(h^{\min(\sigma+\frac{3}{2}-\epsilon,2)})$ in $l^{\infty}$-norm.It's worth mentioning that our spatial error analysis is applicable to highdimensional cube domain by using the properties of tensor product. Moreover,two kinds of averaged schemes are provided to approximate theRiemann--Liouville fractional derivative, and $\mathcal{O}(\tau^{2})$convergence is obtained for all $\alpha\in(0,1)$. Finally, some numericalexperiments verify the effectiveness of the built theory.

Sharp error estimates for spatial-temporal finite difference  approximations to fractional sub-diffusion equation without regularity  assumption on the exact solution