A Stabilized Equal-Order Finite Volume Method for the Stokes Equations

Abstract

We construct a new stabilized finite volume method on rectangular grids for the Stokes equations. The lowest equal-order conforming finite element pair (piecewise bilinear velocities and pressures) and piecewise constant test spaces for both the velocity and pressure are employed in this method. We show the stability of this method and prove first optimal rate of convergence for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm. In addition, a second order optimal error estimate for the velocity in the $L^2$ norm is derived. Numerical experiments illustrating the theoretical results are included.