Estimates of Constants in Poincare Type Inequalities and Applications to A Posteriori Analysis

Integral type inequalities of functional analysis (e.g., Friedrichs, Poincare, trace, LBB inequalities) play an important role in quantitative analysis of PDEs. These constants arise in interpolation type estimates, a posteriori estimates, stability conditions, etc. In the talk we discuss a new method recently suggested for deriving guaranteed bounds of the Friedrichs and Poincare constants in arbitrary polygonal domains. The method is based on ideas of domain decomposition with overlapping subdomains. Numerical tests confirm theoretical results.

Another part of the talk is devoted to estimates of constants in new Poincare type estimates for functions with zero mean boundary traces on a part of the boundary (or on whole boundary). Estimates of the corresponding constants are obtained analytically and by using affine transformations they are extended to a wide collection of basic polygonal domains. Possible applications to mixed type approximations and a posteriori and error estimation methods are discussed.