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 advantages of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries and mesh adaptation. 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：针对任意有界区域上分数阶拉普拉斯算子的数值逼近，提出了一种网格叠加有限差分法。该方法采用非结构单纯形网格与覆盖均匀网格对底层区域进行离散，通过均匀网格有限差分逼近与从非结构网格到均匀网格的数据传输构造近似格式。该方法同时利用均匀网格有限差分逼近在快速傅里叶变换实现高效矩阵-向量乘法方面的优势，以及非结构网格在复杂几何与网格自适应方面的特性。理论分析表明，当数据传输矩阵具有满列秩且列和为正时，其刚度矩阵相似于对称正定矩阵，从而可逆。以分段线性插值作为数据传输的特例进行研究，证明当均匀网格间距小于或等于与非结构网格最小单元高度成比例的正数上界时，线性插值的满列秩与正列和性质得以保证。进一步针对分数阶拉普拉斯算子齐次狄利克雷问题，提出稀疏预条件子用于求解所得线性方程组。数值算例表明，新方法具有与现有有限差分法及有限元法相似的收敛特性，且稀疏预条件技术有效。此外，该方法可便捷地融入现有网格自适应策略，并展示了与MMPDE移动网格方法相结合的数值结果。