A fast algorithm for particle simulations
Journal of Computational Physics
A Generalized Fast Multipole Method for Nonoscillatory Kernels
SIAM Journal on Scientific Computing
A kernel-independent adaptive fast multipole algorithm in two and three dimensions
Journal of Computational Physics
A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
An Accelerated Kernel-Independent Fast Multipole Method in One Dimension
SIAM Journal on Scientific Computing
Superfast Multifrontal Method for Large Structured Linear Systems of Equations
SIAM Journal on Matrix Analysis and Applications
A fast solver for Poisson problems on infinite regular lattices
Journal of Computational and Applied Mathematics
Hi-index | 31.45 |
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(N"b"o"u"n"d"a"r"y+N"s"o"u"r"c"e+N"i"n"c) where N"b"o"u"n"d"a"r"y is the number of nodes on the boundary of the domain, N"s"o"u"r"c"e is the number of nodes subjected to body loads, and N"i"n"c is the number of nodes that deviate from perfect periodicity. This estimate should be compared to the O(N"d"o"m"a"i"nlogN"d"o"m"a"i"n) estimate for FFT based methods, where N"d"o"m"a"i"n is the total number of nodes in the lattice (so that in two dimensions, N"b"o"u"n"d"a"r"y~N"d"o"m"a"i"n^1^/^2). Several numerical examples are presented.