Conforming P ₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes

Abstract

The continuous $P_3$ and discontinuous $P_2$ finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations. By putting two diagonal lines into every quadrilateral of a quadrilateral mesh, we get a subquadrilateral triangular mesh. Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity. Numerical examples show an advantage of such a method over the Taylor-Hood $P_3$-$P_2$ method, where the latter deteriorates when the viscosity becomes small.

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