A unified integral equation scheme for doubly-periodic Laplace and Stokes boundary value problems in two dimensions

摘要

We present a spectrally-accurate scheme to turn a boundary integralformulation for an elliptic PDE on a single unit cell geometry into one for thefully periodic problem. Applications include computing the effectivepermeability of composite media (homogenization), and microfluidic chip design.Our basic idea is to exploit a small least squares solve to apply periodicitywithout ever handling periodic Green's functions. We exhibit fast solvers forthe two-dimensional (2D) doubly-periodic Neumann Laplace problem (flow aroundinsulators), and Stokes non-slip fluid flow problem, that for inclusions withsmooth boundaries achieve 12-digit accuracy, and can handle thousands ofinclusions per unit cell. We split the infinite sum over the lattice of imagesinto a directly-summed "near" part plus a small number of auxiliary sourceswhich represent the (smooth) remaining "far" contribution. Applying physicalboundary conditions on the unit cell walls gives an expanded linear system,which, after a rank-1 or rank-3 correction and a Schur complement, leaves awell-conditioned square system which can be solved iteratively using fastmultipole acceleration plus a low-rank term. We are rather explicit about theconsistency and nullspaces of both the continuous and discretized problems. Thescheme is simple (no lattice sums, Ewald methods, nor particle meshes arerequired), allows adaptivity, and is essentially dimension- andPDE-independent, so would generalize without fuss to 3D and to othernon-oscillatory elliptic problems such as elastostatics. We incorporaterecently developed spectral quadratures that accurately handleclose-to-touching geometries. We include many numerical examples, and provide asoftware implementation.