, Volume 95, Supplement 1, pp 75–88 | Cite as

On optimal node and polynomial degree distribution in one-dimensional \(hp\)-FEM

  • Jan Chleboun
  • Pavel Solin


We are concerned with the task of constructing an optimal higher-order finite element mesh under a constraint on the total number of degrees of freedom. The motivation for this work is to obtain a truly optimal higher-order finite element mesh that can be used to compare the quality of automatic adaptive algorithms. Minimized is the approximation error in a global norm. Optimization variables include the number of elements, positions of nodes, and polynomial degrees of elements. Optimization methods and software that we use are described, and numerical results are presented.


\(hp\)-FEM Optimal mesh Optimal polynomial degree Boundary value problem 

Mathematics Subject Classification (2000)

65K99 65L60 65L10 65L50 



The first author is grateful to Dr. Richard (Dick) Haas for many fruitful discussions.


  1. 1.
    Babuška I, Strouboulis T (2001) The finite element methods and its reliability. Clarendon Press, OxfordzbMATHGoogle Scholar
  2. 2.
    Babuška I, Strouboulis T, Copps K (1997) \(hp\) optimization of finite element approximations: analysis of the optimal mesh sequences in one dimension. Comput Methods Appl Mech Eng 150(1–4):89–108zbMATHCrossRefGoogle Scholar
  3. 3.
    Demkowicz LF (2007) Computing with \(hp\)-adaptive finite elements. Vol. 1: One- and two-dimensional elliptic and Maxwell problems. With CD-ROM. Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  4. 4.
    Demkowicz LF, Kurtz J, Pardo D, Paszyński M, Rachowicz W, Zdunek A (2008) Computing with \(hp\)-adaptive finite elements. Vol. II: Frontiers: three-dimensional elliptic and Maxwell problems with applications. Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  5. 5.
    Dörfler W, Heuveline V (2007) Convergence of an adaptive \(hp\) finite element strategy in one space dimension. Appl Numer Math 57(10):1108–1124MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Eibner T, Melenk J (2007) An adaptive strategy for \(hp\)-FEM based on testing for analyticity. Comput Mech 39(5):575–595MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gui W, Babuška I (1986) The \(h\), \(p\) and \(h\)-\(p\) versions of the finite element method in 1 dimension. III. The adaptive \(h\)-\(p\) version. Numer Math 49:659–683MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Schwab C (1998) \(p\)- and \(hp\)-finite element methods: theory and applications in solid and fluid mechanics. Numerical Mathematics and Scientific Computation. Clarendon Press, OxfordGoogle Scholar
  9. 9.
    Šolín P, Segeth K, Doležel I (2004) Higher-order finite element methods. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  10. 10.
    Wihler TP (2011) An \(hp\)-adaptive strategy based on continuous Sobolev embeddings. J Comput Appl Math 235(8):2731–2739MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Civil EngineeringCzech Technical UniversityPrague 6Czech Republic
  2. 2.Department of Mathematics and StatisticsUniversity of NevadaRenoUSA
  3. 3.Institute of ThermomechanicsPragueCzech Republic

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