  We establish a strategy for finding sharp upper and lower numerical bounds of
the Poincar\'e 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.
