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Nonconforming tetrahedral finite elements for fourth order elliptic equations. (English) Zbl 1125.65105
The aim of this paper is to construct nonconforming tetrahedral finite elements for 3D fourth order elliptic equations. The subject of 2D conforming and nonconforming finite element spaces is wide, but there are few results in the 3D case. This is why this paper is quite interesting. Since conforming elements require high degree polynomials even in the 2D case, attention is paid to nonconforming elements. Two nonconforming tetrahedral finite elements and one quasi-conforming tetrahedral finite element are constructed for the discretization of a fourth order elliptic equation.
More exactly the following finite elements are considered :
– a cubic tetrahedral element with 20 degrees of freedom and complete cubic polynomial shape function space;
– an incomplete cubic tetrahedral element with 16 degrees of freedom and incomplete cubic polynomial shape function space;
– a quasi-conforming tetrahedral element with 16 degrees of freedom similar to a nine-parameter quasi-conforming element; this is a modified Zienkiewicz element.
The convergence of the new elements is demonstrated together with the divergence for the 3D original Zienkiewicz element. Concluding remarks for future work are made.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
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