Computation of sharp estimates of the Poincar\'e constant on planar domains with piecewise self-similar boundary

TL;DR

This paper presents a new numerical method to accurately estimate the Poincaré constant for planar domains with self-similar boundaries, demonstrated on the Koch snowflake.

Contribution

It introduces a novel approach combining shape interpolation, conformal mapping, and spectral problem formulation to compute sharp bounds of the Poincaré constant.

Findings

Successfully computed bounds for the Poincaré constant on the Koch snowflake.

Validated the method's convergence and effectiveness.

Provides a general strategy applicable to similar fractal boundaries.

Abstract

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.