A New Finite Volume Method for the Stokes Problems

Abstract

A new finite volume method for solving the Stokes equations is developed in this paper. The finite volume method makes use of the $BDM_1$ mixed element in approximating the velocity unknown, and consequently, the finite volume solution features a full satisfaction of the divergence-free constraint as required for the exact solution. Optimal-order error estimates are established for the corresponding finite volume solutions in various Sobolev norms. Some preliminary numerical experiments are conducted and presented in the paper. In particular, a post-processing procedure was numerically investigated for the pressure approximation. The result shows a superconvergence for a local averaging post-processing method.