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Poincare´ type inequalities play a central role in the analysis of domain decomposition and multigrid methods for second-order elliptic problems. However, when the coefficient varies within a subdomain or within a coarse grid element, then standard condition number bounds for these methods may be overly pessimistic. In this short note we present new weighted Poincare´ type inequalities for a class of piecewise constant coefficients that lead to sharper bounds independent of any possible large contrasts in the coefficients.
Pechstein, C., & Scheichl, R. (2010). Weighted Poincaré inequalities and applications in domain decomposition. In Domain Decomposition Methods in Science and Engineering XIX (Vol. 78, pp. 197-204). Springer. https://doi.org/10.1007/978-3-642-11304-8_21