A grid-overlay finite difference method is proposed for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying domain and constructs the approximation based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. The method takes full advantage of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Piecewise linear interpolation is studied as a special example for the data transfer. It is proved that the full column rank and positive column sums of linear interpolation is guaranteed if the spacing of the uniform grid is smaller than or equal to a positive bound proportional to the minimum element height of the unstructured mesh. Moreover, a sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem of the fractional Laplacian. Numerical examples demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective. Furthermore, the new method can readily be incorporated with existing mesh adaptation strategies. Numerical results obtained by combining with the so-called MMPDE moving mesh method are also presented.

翻译：提出了一种网格覆盖有限差分方法，用于在任意有界域上数值逼近分数阶拉普拉斯算子。该方法采用非结构化单纯形网格和覆盖均匀网格作为底层域的表示，基于均匀网格有限差分逼近和非结构化网格到均匀网格的数据传递构建近似。该方法充分利用了均匀网格有限差分逼近在通过快速傅里叶变换实现高效矩阵-向量乘法方面的优势，以及非结构化网格处理复杂几何形状的能力。理论分析表明，若数据传递具有满秩列和正列和，则该方法形成的刚度矩阵相似于对称正定矩阵，因而可逆。作为数据传递的特例，研究了分段线性插值方法。证明了当均匀网格间距小于或等于与非结构化网格最小单元高度成正比的正常数上界时，线性插值的满秩列和正列和性质得以保证。此外，针对分数阶拉普拉斯算子齐次狄利克雷问题，提出了稀疏预条件子用于求解所得线性系统的迭代方法。数值实验表明，新方法与现有有限差分和有限元方法具有相似的收敛行为，且稀疏预条件具有有效性。同时，该方法可便捷地融入现有网格自适应策略。文中还展示了与所谓的MMPDE移动网格方法相结合的数值结果。