An unfitted boundary algebraic equation method with Calder\'on preconditioning for 2D Stokes flow in irregular geometry

Abstract

We present an unfitted boundary algebraic equation method for the two-dimensional exterior/interior Stokes equations on a staggered MAC grid. By constructing an explicit free-space pair of velocity and pressure lattice Green's functions (LGFs) from free-space Laplace LGFs, we represent homogeneous fields using sources supported exclusively on thin staggered boundary layers. This formulation imposes physical Dirichlet data at cut points via local interpolation, while sampled-normal rank updates remove hydrostatic null modes associated with single or multiple obstacles. The workflow parallels that of classical boundary integral formulations and requires no artificial boundary conditions for exterior flows, but follows a discretize-then-represent route and does not require singular/near-singular quadrature. The resulting dense boundary system is solved via GMRES, utilizing a componentwise discrete Calder\'on preconditioner built from the scalar Laplace kernel and padded FFTs for fast volume convolutions. Extensive numerical validation, including multiply connected domains, narrow gaps, and Moffatt eddies, confirms discrete incompressibility to solver accuracy and recovers the expected Moffatt eddy scaling. We achieve second-order velocity and pressure convergence and bound maximum discrete divergence within numerical accuracy. The discrete Calder\'on preconditioner reduces the condition number by orders of magnitude and yields nearly mesh-independent conditioning in exterior configurations, while remaining effective---though more demanding---for narrow-gap and fine-grid interior problems.