Sharp bounds of constants in Poincare type inequalities for polygonal domains

TL;DR

This paper derives sharp, easily computable bounds for Poincare inequality constants on polygonal domains, aiding numerical methods and error estimation in PDE solutions.

Contribution

It provides new sharp bounds for Poincare constants on typical numerical domains and demonstrates their application in a posteriori error estimates for elliptic PDEs.

Findings

Sharp bounds for Poincare constants on polygonal domains

Numerical analysis of constant behavior compared with analytical estimates

Application of bounds to improve a posteriori error estimates

Abstract

The paper is concerned with sharp estimates of constants in Poincare type inequalities for functions having zero mean value on the boundary of a Lipschitz domain or on a measurable part of it. These estimates are useful for various numerical methods, in particular, for a posteriori error estimation methods for partial differential equations (PDEs). Therefore, we are mainly focused on domains typical for numerical analysis (simplexes in 2d and 3d) and suggest easily computable relations that provide sharp bounds of the respective constants. Also, we investigate numerically the behavior of the constants in the classical Poincare inequalities and compare these results with known analytical estimates. In the last section, the estimates are used in order to obtain new a posteriori estimates for an elliptic boundary value problem.