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

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Contributions to Partial Differential Equations and Applications

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 47))

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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.

Dedicated to Professor Yuri Kuznetsov on the occasion of his 70th birthday.

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Authors and Affiliations

  1. University of Jyväskylä, P.O. Box 35, 40014, Jyväskylä, Finland

    Sergey Repin

  2. St. Petersburg Department of V.A. Steklov Institute of Mathematics of Russian Academy of Sciences, Saint Petersburg, Russia

    Sergey Repin

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  1. Sergey Repin
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Correspondence to Sergey Repin .

Editor information

Editors and Affiliations

  1. Keldysh Institute of Applied Mathematics (IPM), Moscow, Russia

    B. N. Chetverushkin

  2. College of Technology, University of Houston, Houston, TX, USA

    W. Fitzgibbon

  3. Department of Mathematics, University of Houston, Houston, TX, USA

    Y.A. Kuznetsov

  4. Department of Mathematical Information, University of Jyväskylä, Jyväskylä, Finland

    P. Neittaanmäki

  5. University of Jyväskylä, Jyväskylä, Finland

    J. Periaux

  6. LJLL-UPMC, Paris, France

    O. Pironneau

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Repin, S. (2019). Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods. In: Chetverushkin, B., Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Periaux, J., Pironneau, O. (eds) Contributions to Partial Differential Equations and Applications. Computational Methods in Applied Sciences, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-78325-3_22

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  • DOI: https://doi.org/10.1007/978-3-319-78325-3_22

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-78324-6

  • Online ISBN: 978-3-319-78325-3

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