Weighted Poincare inequalities

Abstract

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 which 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 will study these dependencies in detail. Applications of the inequalities in the analysis of the geometric multigrid, the two-level overlapping Schwarz and the FETI methods can be found in [25, 30].

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

“…Introduced first in [8,22], this approach was rigorously analysed in [7]. The proof relies on uniform (in the coefficients) weighted Poincare inequalities [25]. While this allows full robustness for the small overlap case (cf.…”

Robust Coarsening in Multiscale PDEs

“…The following lemma summarises the results in [14,15,13]. The existence of a benign constant C * T that is independent of α is directly linked to quasimonotonicity, the way in which C * T depends on H T /h to the type.…”

“…It always holds, but in general the constants C * T will not be independent of α| ω T and H T /h. As described in detail in [18, §3] (see also [14,15,13]), to obtain independence of α, we require a certain local quasi-monotonicity of α on each of the regions ω T .…”

New Theoretical Coefficient Robustness Results for FETI-DP

“…The first assumption is a quasi-monotonicity assumption on α. It has been introduced in Dryja et al [1996] and generalized in Klawonn et al [2002], Pechstein and Scheichl [2010]. The second assumption states that X "sees" the largest coefficient.…”