Fractional discrete Laplacian versus discretized fractional Laplacian
This work provides a rigorous mathematical foundation for fractional discrete Laplacians, which is important for numerical analysis and PDEs, but the results are incremental and domain-specific.
The paper defines the fractional discrete Laplacian and compares it to the discretized continuous fractional Laplacian, deriving error estimates in ℓ∞ as h→0 under minimal regularity assumptions. It also provides a pointwise kernel formula and Hölder estimates for the discrete operator.
We define and study some properties of the fractional powers of the discrete Laplacian $$(-Δ_h)^s,\quad\hbox{on}~\mathbb{Z}_h = h\mathbb{Z},$$ for $h>0$ and $0<s<1$. A comparison between our fractional discrete Laplacian and the \textit{discretized} continuous fractional Laplacian as $h\to0$ is carried out. We get estimates in $\ell^\infty$ for the error of the approximation in terms of $h$ under minimal regularity assumptions. Moreover, we provide a pointwise formula with an explicit kernel and deduce Hölder estimates for $(-Δ_h)^s$. A study of the negative powers (or discrete fractional integral) $(-Δ_h)^{-s}$ is also sketched. Our analysis is mainly performed in dimension one. Nevertheless, we show certain asymptotic estimates for the kernel in dimension two that can be extended to higher dimensions. Some examples are plotted to illustrate the comparison in both one and two dimensions.