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

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


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.


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


  1. 1.
    Acosta G, Durán RG (2004) An optimal Poincaré inequality in \(L^1\) for convex domains. Proc Amer Math Soc 132(1):195–202MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnold D, Boffi D, Falk R (2002) Approximation by quadrilateral finite elements. Math Comp 71(239):909–922MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arnold D, Boffi D, Falk R (2005) Quadrilateral H(div) finite elements. SIAM J Numer Anal 42(6):2429–2451MathSciNetCrossRefGoogle Scholar
  4. 4.
    Babuška I, Aziz A (1976) On the angle condition in the finite element method. SIAM J Numer Anal 13(2):214–226MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bermúdez A, Gamallo P, Nogueiras MR, Rodríguez R (2005) Approximation properties of lowest-order hexahedral Raviart-Thomas finite elements. C R Math Acad Sci Paris 340(9):687–692MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New YorkCrossRefGoogle Scholar
  7. 7.
    Brezzi F, Lipnikov K, Shashkov M, Simoncini V (2007) A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput Methods Appl Mech Engrg 196(37–40):3682–3692MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cheng SY (1975) Eigenvalue comparison theorems and its geometric applications. Math Z 143(3):289–297MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chua S-K, Wheeden RL (2006) Estimates of best constants for weighted Poincaré inequalities on convex domains. Proc London Math Soc (3), 93(1):197–226Google Scholar
  10. 10.
    Chua S-K, Wheeden RL (2010) Weighted Poincaré inequalities on convex domains. Math Res Lett 17(5):993–1011MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fox DW, Kuttler JR (1983) Sloshing frequencies. Z Angew Math Phys 34(5):668–696MathSciNetCrossRefGoogle Scholar
  12. 12.
    Girault V, Raviart PA (1986) Finite element methods for Navier-Stokes equations: theory and algorithms. Springer, BerlinCrossRefGoogle Scholar
  13. 13.
    Hackbusch W, Löhndorf M, Sauter SA (2006) Coarsening of boundary-element spaces. Computing 77(3):253–273MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hecht F (2012) New development in FreeFem++. J Numer Math 20(3–4):251–265MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kozlov V, Kuznetsov N (2004) The ice-fishing problem: The fundamental sloshing frequency versus geometry of holes. Math Methods Appl Sci 27(3):289–312MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kozlov V, Kuznetsov N, Motygin O (2004) On the two-dimensional sloshing problem. Proc R Soc Lond Ser A Math Phys Eng Sci 460(2049):2587–2603MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuznetsov Yu (2006) Mixed finite element method for diffusion equations on polygonal meshes with mixed cells. J Numer Math 14(4):305–315MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kuznetsov Yu (2011) Approximations with piece-wise constant fluxes for diffusion equations. J Numer Math 19(4):309–328MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kuznetsov Yu (2014) Mixed FE method with piece-wise constant fluxes on polyhedral meshes. Russian J Numer Anal Math Modelling 29(4):231–237MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kuznetsov Yu (2015) Error estimates for the \(RT_0\) and PWCF methods for the diffusion equations on triangular and tetrahedral meshes. Russian J Numer Anal Math Modelling 30(2):95–102MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kuznetsov Yu, Prokopenko A (2010) A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous media. Numer Linear Algebra Appl 17(5):759–769MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kuznetsov Yu, Repin S (2003) New mixed finite element method on polygonal and polyhedral meshes. Russian J Numer Anal Math Modelling 18(3):261–278MathSciNetCrossRefGoogle Scholar
  23. 23.
    Laugesen RS, Siudeja BA (2010) Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality. J Differen Equat 249(1):118–135CrossRefGoogle Scholar
  24. 24.
    Mali O, Neittaanmäki P, Repin S (2014) Accuracy verification methods: Theory and algorithms, vol 32. Computational Methods in Applied Sciences. Springer, DordrechtGoogle Scholar
  25. 25.
    Matculevich S, Neittaanmäki P, Repin S (2015) A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne-Weinberger inequality. Discrete Contin Dyn Syst 35(6):2659–2677MathSciNetCrossRefGoogle Scholar
  26. 26.
    Matculevich S, Repin S (2016) Explicit constants in Poincaré-type inequalities for simplicial domains and application to a posteriori estimates. Comput Methods Appl Math 16(2):277–298MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nazarov A, Repin S (2015) Exact constants in Poincaré type inequalities for functions with zero mean boundary traces. Math Methods Appl Sci 38(15):3195–3207MathSciNetCrossRefGoogle Scholar
  28. 28.
    Payne LE, Weinberger HF (1960) An optimal Poincaré inequality for convex domains. Arch Rational Mech Anal 5:286–292MathSciNetCrossRefGoogle Scholar
  29. 29.
    Poincaré H (1894) Sur les équations de la physique mathématique. Rend Circ Mat Palermo 8:57–155CrossRefGoogle Scholar
  30. 30.
    Repin S (2008) A posteriori estimates for partial differential equations. Walter de Gruyter, BerlinCrossRefGoogle Scholar
  31. 31.
    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 – ENUMATH2013, volume 103 of Lecture Notes in Computational Science and Engineering, pp 215–223Google Scholar
  32. 32.
    Repin S (2015) Interpolation of functions based on Poincaré type inequalities for functions with zero mean boundary traces. Russian J Numer Anal Math Modelling 30(2):111–120MathSciNetCrossRefGoogle Scholar
  33. 33.
    Roberts JE, Thomas J-M (1991) Mixed and hybrid methods. In: Handbook of Numerical Analysis, Vol II, pp 523–639. North-Holland, Amsterdam,Google Scholar
  34. 34.
    Steklov VA (1896) On the expansion of a given function into a series of harmonic functions. Commun Kharkov Math Soc Ser 2(5):60–73 (in Russian)Google Scholar

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

Personalised recommendations