A fast multiresolution lattice Greens function method for elliptic difference equations Abstract   We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Greenu0027s 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 timetosolution. We extend this method to a multiresolution 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. Secondorder accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost.

Solving Elliptic Interface Problems with Jump Conditions on Cartesian Grids Abstract   We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces secondorder accurate solutions and firstorder accurate gradients in the      L    xe2x88x9e     norm on Cartesian grids. The condition number is bounded, regardless of the ratio of the diffusion constant and scales like that of the standard 5point stencil approximation on a rectangular grid with no interface. Numerical examples are given in two and three spatial dimensions.