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

Source abstract

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