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Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods

  • Sergey RepinEmail author
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)

Abstract

We consider inequalities of the Poincaré–Steklov type for subspaces of \(H^1\)-functions defined in a bounded domain \(\varOmega \in \mathbb {R}^d\) with Lipschitz boundary \(\partial \varOmega \). For scalar valued functions, the subspaces are defined by zero mean condition on \(\partial \varOmega \) or on a part of \(\partial \varOmega \) having positive \(d-1\) measure. For vector valued functions, zero mean conditions are applied to normal components on plane faces of \(\partial \varOmega \) (or to averaged normal components on curvilinear faces). We find explicit and simply computable bounds of constants in the respective Poincaré type inequalities for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second part of the paper discusses applications of the estimates to interpolation of scalar and vector valued functions on macrocells and on meshes with non-overlapping and overlapping cells.

Keywords

Poincaré type inequalities Interpolation of functions Estimates of constants in functional inequalities 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of JyväskyläJyväskyläFinland
  2. 2.St. Petersburg Department of V.A. Steklov Institute of Mathematics of Russian Academy of SciencesSaint PetersburgRussia

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