Numerische Mathematik

, Volume 35, Issue 3, pp 315–341

Mixed finite elements in ℝ3


  • J. C. Nedelec
    • Centre de Mathématiques Appliquées-Ecole Polytechnique
Sign-Stability in Difference Schemes for Parabolic Initinal-Boundary Value Problems

DOI: 10.1007/BF01396415

Cite this article as:
Nedelec, J.C. Numer. Math. (1980) 35: 315. doi:10.1007/BF01396415


We present here some new families of non conforming finite elements in ℝ3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.

Subject Classifications

AMS(MOS) 65N30 CR: 5.17

First, we introduce some notations


is a tetrahedron or a cube, thevolume of which is\(\int\limits_K^{} {dx}\)


is its boundary


is a face ofK, thesurface of which is\(\int\limits_f^{} {dy}\)


is an edge, the length of which is\(\int\limits_a^{} {ds}\)

L 2 (K)

is the usual Hilbert space of square integrable functions defined onK

H m (K)

{Φ∈L 2(K); ∂αΦ∈L 2(K); |α|≦m}, where α=(α1, α2, α3) is a multi-index; |α|=α123


∇∧u, (defined by using the distributional derivative) foru=(u 1,u 2,u 3);u iL 2 (K)


{u∈(L 2 (K))3; curlu∈(L 2 (K)) 3}




{u∈(L 2 (K)) 3; divuL 2 (K)}

D k u

is thek-th differential operator associated tou, which is a (k+1)-multilinear operator acting on ℝ3


is an index


is the linear space of polynomials, the degree of which is less or equal tok

σ k

is the group of all permutations of the set {1, 2, ...,k}

c orc ε

will stand for any constant depending possibly on ε

Copyright information

© Springer-Verlag 1980