We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature, which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector products. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.

翻译：本文提出一种基于半离散傅里叶变换的新型简洁谱方法，用于离散分数阶拉普拉斯算子$(-\Delta)^\frac{\alpha}{2}$。通过数值分析和实验验证其性能。该方法在离散层面与分数阶拉普拉斯算子$(-\Delta)^\frac{\alpha}{2}$具有相同符号$|\xi|^\alpha$，因此可视为分数阶拉普拉斯算子的精确离散模拟。这一独特特性使本方法与现有其他分数阶拉普拉斯方法存在本质区别。值得注意的是，本方法不同于文献中通常局限于周期边界条件的傅里叶拟谱方法（见注记\ref{remark0}）。数值分析表明该方法可实现谱精度，并分析了求解分数阶泊松方程时的稳定性与收敛性。所提方案生成多层托普利兹刚度矩阵，因此可开发快速算法实现高效矩阵-向量乘积。计算复杂度为${\mathcal O}(2N\log(2N))$，内存存储量为${\mathcal O}(N)$（$N$为总网格点数）。大量数值实验验证了理论分析结果，并展示了该方法求解各类问题的有效性。