Analysis of the stokes problem in a regular bounded open set

Abstract

The Stokes equation describes the flow velocity of a steady state fluid in relation to the pressure and the external source. The corresponding variational formulation of the Stokes equation is studied in the paper. In more detail, we delve into the analysis of the equivalence relations pertaining to the variational formulations of the Stokes equations. We found that the variational formulation of the Stokes equation can be approximated by a type of variational formulation with a coefficient but without the constraint on the divergence. Then we did a analysis on the approximation by the finite element method to the the variational formulation without the constraint on the divergence, and we find that we should use preconditioning techniques before using the iteration. More precisely, we give an error analysis of this numerical computation method through rigorous proofs, and from this we deduce the need to use preprocessing techniques to avoid long computation times.