A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

Abstract

We present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green's functions. We describe fast solvers for the two-dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed "near" part plus a small number of auxiliary sources that represent the (smooth) remaining "far" contribution. Applying physical boundary 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 a well-conditioned square system that can be solved iteratively using fast multipole acceleration plus a low-rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension-and PDE-independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close-to-touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against highaccuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 10 4 randomly located inclusions per unit cell.

Flexibly imposing periodicity in kernel independent FMM: A multipole-to-local operator approach

“…Gumerov and Duraiswami used a similar formulation in 3D, except that the periodicity is checked for points on a spherical (in 3D) surface instead of on the L, R, T, B boundaries of the domain. Similar uniqueness argument is also proved in the work by Barnett et al [11] for doubly periodic Stokes problems on 2D domains. When the above method is extended to partial periodicity, uniqueness is no longer guaranteed by simply imposing the periodic boundary condition.…”

“…When the aspect ratio of B 0 is of order (1), Ewald methods can still be used. For higher but still moderate aspect ratios, Ewald methods may converge slowly but we can utilize the scheme in [11] to construct the periodic kernel K P directly without a series summation, with some tuning of the discretization of the check and equivalent surfaces to fit the high aspect ratio. For extremely high aspect ratio the accurate construction of the periodic kernel K P remains to be studied.…”

“…Barnett et al [11] showed that the solution to this equation is unique up to a constant. This constant can be later determined by some other physical information of the problem, like a fixed temperature at some point.…”

“…For the biharmonic equation in 3D space, a series summation in Fourier space is also sometimes sufficient because the Green's function converges as 1/k 4 in Fourier space. If K P is not easy to find analytically, the method of Barnett et al [11] can be used if the uniqueness is guaranteed as discussed above. The Hierarchical Repetition method in [9] is also applicable if the far-field is properly fixed to satisfy the physical compatibility condition.…”

A boundary-integral framework to simulate viscous erosion of a porous medium

“…For the porous network simulations considered here, it is essential to resolve nonlocal effects between the different bodies. BIEs naturally resolve these interactions, and they have been used extensively to simulate viscous fluids in complex geometries, including porous media flow [4,22]. Common challenges of a BIE formulation include developing efficient preconditioners for the discretized system and numerical difficulties associated with nearly touching bodies.…”