Divergence-free finite elements on tetrahedral grids for k≥6

It was shown two decades ago that the Pk-Pk−1 mixed element on triangular grids, approximating the velocity by the continuous Pk piecewise polynomials and the pressure by the discontinuous Pk−1 piecewise polynomials, is stable for all k ≥ 4, provided the grids are free of a nearly-singular vertex. The problem with the method in 3D was posted then and remains open. The problem is solved partially in this work. It is shown that the PkPk−1 element is stable and of optimal order in approximation, on a family of uniform tetrahedral grids, for all k ≥ 6. The analysis is to be generalized to non-uniform grids, when we can deal with the complicity of 3D geometry. For the divergence-free elements, the finite element spaces for the pressure can be avoided in computation, if a classic iterated penalty method is applied. The finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity. Numerical tests are provided.