We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast multipole method in conjunction with fast Fourier transforms to yield linear complexity and decrease time-to-solution. We extend this method to a multi-resolution scheme and allow for locally refined Cartesian blocks embedded in the computational domain. Appropriately chosen interpolation and regularization operators retain consistency between the discrete Laplace operator and its inverse on the unbounded domain. Second-order accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost.

我们提出了一种网格细化技术，用于基于快速格子格林函数（FLGF）方法求解无界域上的椭圆差分方程。FLGF方法利用笛卡尔网格的规则性，并结合使用快速多极方法和快速傅立叶变换来产生线性复杂度并减少求解时间。我们将此方法扩展到一个多分辨率方案，并允许在计算域中嵌入本地精炼的笛卡尔块。适当地选择内插和正规化运营商保留离散拉普拉斯算子和其上无界域逆之间的一致性。保持了二阶精度和线性复杂度，同时显着减少了自由度的数量，从而减少了计算成本。