Not exactly a mistake, but an oversight in the derivation. One would expect the set of 18 shape functions would be invariant under exchange of nodes, but this is not the case. Factor the shape functions and you will see the inconsistencies. Also see the lack of symmetry in the last entry in the list of the space spanned. The element can be defined with only four shapes, the scalar, first derivative wrt x (or y), the second derivative wrt xx (or yy) and derivative wrt xy at node (0,0). The remaining 14 can be found from symmetry and coordinate transformations.
My interest is not so much in this element, but rather in divergence-free vector-valued elements. In 2D, with Hermite elements like the Bell, interchange the two first derivative functions and change the sign on the appropriate one and you have a scalar element with stream function and div-free velocity DOFs. Take the curl and you have the div-free finite element. Of course the element must have normal continuity along the edges, but this will be the case for C0 elements. 3D is a little more complicated. Anyway, I can supply corrected form if anyone is interested. Over 65 years uncorrected - maybe no one uses it.
Not exactly a mistake, but an oversight in the derivation. One would expect the set of 18 shape functions would be invariant under exchange of nodes, but this is not the case. Factor the shape functions and you will see the inconsistencies. Also see the lack of symmetry in the last entry in the list of the space spanned. The element can be defined with only four shapes, the scalar, first derivative wrt x (or y), the second derivative wrt xx (or yy) and derivative wrt xy at node (0,0). The remaining 14 can be found from symmetry and coordinate transformations.
My interest is not so much in this element, but rather in divergence-free vector-valued elements. In 2D, with Hermite elements like the Bell, interchange the two first derivative functions and change the sign on the appropriate one and you have a scalar element with stream function and div-free velocity DOFs. Take the curl and you have the div-free finite element. Of course the element must have normal continuity along the edges, but this will be the case for C0 elements. 3D is a little more complicated. Anyway, I can supply corrected form if anyone is interested. Over 65 years uncorrected - maybe no one uses it.