Weighted Poincaré inequalities

Poincaré-type inequalities are a key tool in the analysis of partial differential equations. They play a particularly central role in the analysis of domain decomposition and multilevel iterative methods for second-order elliptic problems. When the diffusion coefficient varies within a subdomain or within a coarse grid element, then condition number bounds for these methods based on standard Poincaré inequalities may be overly pessimistic. In this paper, we present new results on weighted Poincaré-type inequalities for very general classes of coefficients that lead to sharper bounds independent of any possible large variation in the coefficients. The main requirement on the coefficients is some form of quasi-monotonicity that we will carefully describe and analyse. The Poincaré constants depend on the topology and the geometry of regions of relatively high and/or low coefficient values, and we shall study these dependencies in detail. Applications of the inequalities in the analysis of domain decomposition and multigrid methods can be found in Pechstein & Scheichl (2011, Numer. Math., 118) and Scheichl et al. (2012, SIAM J. Numer. Anal., 50).