Foundations of Computational Mathematics

, Volume 11, Issue 3, pp 337–344 | Cite as

The Serendipity Family of Finite Elements



We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least sr of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r−2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.


Serendipity Finite element Unisolvence 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T.J.R. Hughes, The Finite Element Method (Englewood Cliffs, Prentice-Hall, 1987). Linear static and dynamic finite element analysis, with the collaboration of Robert M. Ferencz and Arthur M. Raefsky. MATHGoogle Scholar
  2. 2.
    V.N. Kaliakin, Introduction to Approximate Solution Techniques, Numerical Modeling, & Finite Element Methods (CRC Press, Boca Raton, 2001). Civil and Environmental Engineering. Google Scholar
  3. 3.
    J. Mandel, Iterative solvers by substructuring for the p-version finite element method, Comput. Methods Appl. Mech. Eng. 80(1–3), 117–128 (1990). Spectral and high order methods for partial differential equations (Como, 1989). MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    G. Strang, G.J. Fix, An analysis of the finite element method, in Prentice-Hall Series in Automatic Computation (Prentice-Hall, Englewood Cliffs, 1973). Google Scholar
  5. 5.
    B. Szabó, I. Babuška, Finite Element Analysis (Wiley-Interscience, New York, 1991). MATHGoogle Scholar
  6. 6.
    O.C. Zinkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th edn., vol. 1 (Butterworth, Stoneham, 2005). Google Scholar

Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDekalbUSA

Personalised recommendations