Analysis of a Streamline Diffusion Finite Element Method for the Stokes and Navier–Stokes Equations

Abstract

For the Stokes equations with convection and the incompressible Navier–Stokes equations, the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure, thus allowing the use of arbitrary pairs of velocity-pressure spaces. For the linear problem, the authors obtain for all mesh–Peclet numbers optimal error estimates in natural norms including, in particular, the $L^2 $-norm of the pressure. The same holds for the nonlinear problem, which close to a regular branch of solutions, i.e., the linearized operator, is an isomorphism of the norm of the inverse of which still depends on the Reynolds number. Consequently, the dependence of the error constants on the Reynolds number is not completely resolved in this case.