Estimates of Constants in Boundary-Mean Trace Inequalities and Applications to Error Analysis
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We discuss Poincaré type inequalities for functions with zero mean values on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. For some basic domains (rectangles, quadrilaterals, and right triangles) exact constants in these inequalities has been found in Nazarov and Repin (ArXiv Ser Math Anal, 2012, arXiv:1211.2224). We shortly discuss two examples, which show that the estimates can be helpful for quantitative analysis of PDEs. In the first example, we deduce estimates of modeling errors generated by simplification (coarsening) of a boundary value problem. The second example presents a new form of the functional type a posteriori estimate that provides fully guaranteed and computable bounds of approximation errors. Constants in Poincaré type inequalities enter these estimates.
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