Neilan's divergence‐free finite elements for Stokes equations on tetrahedral grids

Abstract

The Neilan Pk$$ {P}_k $$ ‐ Pk−1$$ {P}_{k-1} $$ divergence‐free finite element is stable on any tetrahedral grid, where the piece‐wise Pk$$ {P}_k $$ polynomial velocity is C0$$ {C}^0 $$ on the grid, C1$$ {C}^1 $$ on edges and C2$$ {C}^2 $$ at vertices, and the piece‐wise Pk−1$$ {P}_{k-1} $$ polynomial pressure is C0$$ {C}^0 $$ on edges and C1$$ {C}^1 $$ at vertices. However the method does not work if the exact pressure solution does not vanish on all domain edges, because of the excessive continuity requirements. We extend the Neilan element by removing the extra requirements at domain boundary edges. That is, if a vertex is on a domain boundary edge and if an edge has one endpoint on a domain boundary edge, the velocity is only C0$$ {C}^0 $$ at the vertex and on the edge, respectively, and the pressure is totally discontinuous there. Under the condition that no tetrahedron in the grid has more than one face‐triangle on the domain boundary, we prove that the extended finite element is stable, and consequently produces solutions of optimal order convergence for all Stokes problems. A numerical example is given, confirming the theory.