A family of 3D continuously differentiable finite elements on tetrahedral grids

A family of continuously differentiable piecewise polynomials of degree 9 and higher, on general tetrahedral grids, is constructed, by simplifying and extending the P9 element of Ženišek. A mathematical justification and numerical tests are presented.The current computing power is still limited for the computation with 3D C1 finite elements in general. The construction here mainly serves the purposes of understanding and ensuring the approximation properties of C1 finite elements spaces on tetrahedral grids. In particular, this construction indicates that the 3D divergence-free C0-Pk elements have the full order of approximation for any degree k⩾8.