Fast and accurate numerical methods for solving elliptic difference equations defined on lattices

Techniques for rapidly computing approximate solutions to elliptic PDEs such as Laplace’s equation are well established. For problems involving general domains, and operators with constant coefficients, a highly efficient approach is to rewrite the boundary value problem as a Boundary Integral Equation (BIE), and then solve the BIE using fast methods such as, e.g., the Fast Multipole Method (FMM). The current paper demonstrates that this procedure can be extended to elliptic difference equations defined on infinite lattices, or on finite lattice with boundary conditions of either Dirichlet or Neumann type. As a representative model problem, a lattice equivalent of Laplace’s equation on a square lattice in two dimensions is considered: discrete analogs of BIEs are derived and fast solvers analogous to the FMM are constructed. Fast techniques are also constructed for problems involving lattices with inclusions and local deviations from perfect periodicity. The complexity of the methods described is O(Nboundary + Nsource + Ninc) where Nboundary is the number of nodes on the boundary of the domain, Nsource is the number of nodes subjected to body loads, and Ninc is the number of nodes that deviate from perfect periodicity. This estimate should be compared to the O(NdomainlogNdomain) estimate for FFT based methods, where Ndomain is the total number of nodes in the lattice (so that in two dimensions, N boundary ∼ N domain 1 / 2 ). Several numerical examples are presented.