Nonstandard Nonconforming Approximation of the Stokes Problem, I: Periodic Boundary Conditions

Abstract

This paper analyzes a nonstandard form of the Stokes problem where the mass conservation equation is expressed in the form of a Poisson equation for the pressure. This problem is shown to be well-posed in the $d$-dimensional torus. A nonconforming approximation is proposed and, contrary to what happens when using the standard saddle-point formulation, the proposed setting is shown to yield optimal convergence for every pairs of approximation spaces.