Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22(4), 751–756 (2003)
C.R. Dohrmann, A. Klawonn, O.B. Widlund, Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)
V.G. Maz’ja, Classes of domains and imbedding theorems for function spaces. Sov. Math. Dokl. 1, 882–885 (1960)
A. Nazarov, S. Repin, Exact constants in Poincaré type inequalities for functions with zero mean boundary traces. ArXiv Ser. Math. Anal. (2012). arXiv:1211.2224
S. Nicaise, A posteriori error estimations of some cell-centered finite volume methods. SIAM J. Numer. Anal. 43(4), 1481–1503 (2005)
L.E. Payne, H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
G.V. Pencheva, M. Vohralik, M.F. Wheeler, T. Wildey, Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51(1), 526–554 (2013)
H. Poincaré, Sur les équations aux deŕiveés partielles de la physique mathematique. Am. J. Math. 12, 211–294 (1890)
H. Poincaré, Sur les équations de la physique mathematique. Rend. Circ. Mat. Palermo, 8, 57–156 (1894)
S. Repin, A Posteriori Error Estimates for Partial Differential Equations (Walter de Gruyter, Berlin, 2008)
M.F. Wheeler, I. Yotov, A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43(3), 1021–1042 (2005)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Repin, S. (2015). Estimates of Constants in Boundary-Mean Trace Inequalities and Applications to Error Analysis. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-10705-9_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10704-2
Online ISBN: 978-3-319-10705-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)