Computation of sharp estimates of the Poincaré constant on planar domains with piecewise self-similar boundary

We establish a strategy for finding sharp upper and lower numerical bounds of the Poincaré constant on a class of planar domains with piecewise self-similar boundary. The approach consists of four main components: W1) tight inner-outer shape interpolation, W2) conformal mapping of the approximate polygonal regions, W3) grad-div system formulation of the spectral problem and W4) computation of the eigenvalue bounds. After describing the method, justifying its validity and determining general convergence estimates, we show concrete evidence of its effectiveness by computing lower and upper bound estimates for the constant on the Koch snowflake.