Finite element approximation for the fractional eigenvalue problem

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.Comment: 20 pages, 6 figure

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u \in C^{\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^\epsilon)$, with $\epsilon > 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems.Comment: 24 pages, 6 figures, and 6 table

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

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.Comment: 21 pages, 7 figure