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QUADRATIC VELOCITY/LINEAR PRESSURE STOKES ELEMENTS* DOUGLAS N. ARNOLD and JINSHUI QIN
 

Summary: QUADRATIC VELOCITY/LINEAR PRESSURE STOKES ELEMENTS*
DOUGLAS N. ARNOLD and JINSHUI QIN
Abstract. We study the finite element approximation of the stationary Stokes equations in the velocity-
pressure formulation using continuous piecewise quadratic functions for velocity and discontinuous piece-
wise linear functions for pressure. For some meshes this method is unstable, even after spurious pressure
modes are removed. For other meshes there are spurious local pressure modes, but once they are removed
the method is stable, and in particular, the velocity converges with optimal order. On yet other meshes
there are no spurious pressure modes and the method is stable and optimally convergent for both pressure
and velocity.
1. Introduction. We study the approximation of the Stokes equations with no-slip
boundary conditions:
- u + p = f in , div u = 0 in , u = 0 on . (1)
A finite element discretization of (1) is based on the weak formulation which seeks (u, p)
V P := H1
() L2
() such that
a(u, v) - b(v, p) = (f, v) v V , b(u, q) = 0 q P,
where
a(u, v) :=

  

Source: Arnold, Douglas N. - School of Mathematics, University of Minnesota

 

Collections: Mathematics