Numerische Mathematik

, Volume 35, Issue 3, pp 315–341

# Mixed finite elements in ℝ3

• J. C. Nedelec
Sign-Stability in Difference Schemes for Parabolic Initinal-Boundary Value Problems

DOI: 10.1007/BF01396415

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

## Summary

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

K

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

K

is its boundary

f

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

a

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

L2(K)

is the usual Hilbert space of square integrable functions defined onK

Hm(K)

{Φ∈L2(K); ∂αΦ∈L2(K); |α|≦m}, where α=(α1, α2, α3) is a multi-index; |α|=α123

curlu

∇∧u, (defined by using the distributional derivative) foru=(u1,u2,u3);uiL2(K)

H(curl)

{u∈(L2(K))3; curlu∈(L2(K))3}

divu

∇·u

H(div)

{u∈(L2(K))3; divuL2(K)}

Dku

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

k

is an index

k

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

σk

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

c orcε

will stand for any constant depending possibly on ε