Sharp bounds of constants in Poincare type inequalities for polygonal domains
For numerical analysts, this work offers practical sharp bounds for constants used in a posteriori error estimation, though the results are incremental improvements over known analytical estimates.
The paper provides sharp bounds for constants in Poincaré-type inequalities on polygonal domains, with easily computable relations for simplexes in 2D and 3D. These bounds are applied to obtain new a posteriori error estimates for elliptic boundary value problems.
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.