Poisson equations appear in many graphics settings including, but not limited to, physics-based fluid simulation. Numerical solvers for such problems strike context-specific memory, performance, stability and accuracy trade-offs. We propose a new Poisson filter-based solver that balances between the strengths of spectral and iterative methods. We derive universal Poisson kernels for forward and inverse Poisson problems, leveraging careful adaptive filter truncation to localize their extent, all while maintaining stability and accuracy. Iterative composition of our compact filters improves solver iteration time by orders-of-magnitude compared to optimized linear methods. While motivated by spectral formulations, we overcome important limitations of spectral methods while retaining many of their desirable properties. We focus on the application of our method to high-performance and high-fidelity fluid simulation, but we also demonstrate its broader applicability.