Efficient Open-Boundary Poisson Solves

Poisson equations arise pervasively in computer graphics, yet efficiently solving them on unbounded domains remains a long-standing challenge. Existing approaches attempting to model boundary conditions at infinity either incur prohibitive computational costs by expanding the simulation domain, sacrifice accuracy through artificial boundary conditions, or lack the flexibility to handle varying coefficients. In this paper, we introduce a simple and practical method for solving generalized Poisson equations on finite domains with open boundaries, enabling exact enforcement of asymptotic conditions at infinity. Our method couples a finite-volume interior solver with a boundary-element formulation of the exterior problem, enforcing continuity of both the solution and its normal derivative across an artificial boundary. Inspired by Johnson-Nédélec coupling and interface relaxation, we develop a partitioned, iterative scheme based on damped fixed-point iterations, which leverages fast, inexact boundary-element solves for the exterior problem and efficient sparse solves for the interior. Theoretical analysis and empirical tests confirm that our method provides superior accuracy and performance compared to existing ad-hoc solutions across a variety of graphics applications.