Divergence-free finite elements on tetrahedral grids for $k\ge6$

Abstract

It was shown two decades ago that the $P_k$-$P_{k-1}$ mixed element on triangular grids, approximating the velocity by the continuous $P_k$ piecewise polynomials and the pressure by the discontinuous $P_{k-1}$ piecewise polynomials, is stable for all $k\ge 4$, 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 $P_k$-$P_{k-1}$ element is stable and of optimal order in approximation, on a family of uniform tetrahedral grids, for all $k\ge 6$. 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.