In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ for $\alpha \in (0, 2)$. The main advantage of our method is to easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain the scheme structure and computer implementation unchanged. Moreover, our discretization of the fractional Laplacian results in a symmetric (multilevel) Toeplitz differentiation matrix, which not only saves memory cost in simulations but enables efficient computations via the fast Fourier transforms. The performance of our method in both approximating the fractional Laplacian and solving the fractional Poisson problems was detailedly examined. It shows that our method has an optimal accuracy of ${\mathcal O}(h^2)$ for constant or linear basis functions, while ${\mathcal O}(h^4)$ if quadratic basis functions are used, with $h$ a small mesh size. Note that this accuracy holds for any $\alpha \in (0, 2)$ and can be further increased if higher-degree basis functions are used. If the solution of fractional Poisson problem satisfies $u \in C^{m, l}(\bar{\Omega})$ for $m \in {\mathbb N}$ and $0 < l < 1$, then our method has an accuracy of ${\mathcal O}\big(h^{\min\{m+l,\, 2\}}\big)$ for constant and linear basis functions, while ${\mathcal O}\big(h^{\min\{m+l,\, 4\}}\big)$ for quadratic basis functions. Additionally, our method can be readily applied to study generalized fractional Laplacians with a symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.
翻译:在本文中, 我们提出一个新的操作器因子化方法, 以分解分解分解分解( delta) $ (- Delta) {frac_ alpha}2} 美元 。 我们方法的主要优点是使用高度拉格朗基函数, 很容易提高数字准确性, 但仍保持方案结构和计算机实施 。 此外, 我们分解分解的拉格化方法导致对称( 多级) 托普利茨 差分解矩阵, 它不仅在模拟中节省记忆成本, 而且能够通过快速的 Fleier 变换来高效计算 。 我们方法在接近分解的拉格基和解决分解问题时的性能得到了详细检查。 这表明我们的方法在恒定或线基函数中具有最佳的精度值 $ ( h) 。 如果使用复数基函数, 以小量的 $ (h) (h) (h) (h) 平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平方平。