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.
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Abbreviations
- 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\(\int\limits_a^{} {ds}\)
- 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|=α1+α2+α3
- curlu :
-
∇∧u, (defined by using the distributional derivative) foru=(u 1,u 2,u 3);u i∈L 2 (K)
- H(curl):
-
{u∈(L 2 (K))3; curlu∈(L 2 (K)) 3}
- divu :
-
∇·u
- H(div):
-
{u∈(L 2 (K)) 3; divu∈L 2 (K)}
- D k u :
-
is thek-th differential operator associated tou, which is a (k+1)-multilinear operator acting on ℝ3
- k :
-
is an index
- ℙ k :
-
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
References
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