A P4 bubble enriched P3 divergence-free finite element on triangular grids

On triangular grids, the continuous Pk plus discontinuous Pk−1 mixed finite element is stable for polynomial degree k≥4. When k=3, the inf-sup condition fails and the mixed finite element converges at an order that is two orders lower than the optimal order. We enrich the continuous P3 by adding some P4 divergence-free bubble functions, to be exact, one P4 divergence-free bubble function each component each edge. We show that such an enriched P3-P2 mixed element is inf-sup stable, and converges at the optimal order. Numerical tests are presented, comparing the new element with the P4-P3 element and the unstable P3-P2 element.