A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

“…The upwind version of the Crouzeix-Raviart finite element method was developed and analyzed in [37] for a linear stationary convection-diffusion equation. This was the inspiration for Schieweck and Tobiska who investigated in [40] upwind schemes for steady Navier-Stokes equations. In [2] the convergence analysis of the combined barycentric finite volume-nonconforming finite element method applied to a nonlinear convection-diffusion problem is given.…”

“…There is an extensive literature on the numerical solution of convection-diffusion problems. Let us mention, e.g., the papers [1], [25], [26], [37], [40], [46], [48], [49] and the monographs [36], [39] (and the references therein), devoted mainly to linear problems. The main difficulty which must be overcome is the precise resolution of the so-called boundary layers.…”

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

“…B, is supposed to be the barycenter of the face r,. Now we can define the discrete spaces V,, = V and Q,, = Q by ( 5 ) where f,,,(K),m = 0, I , is the set of all polynomials on K with degree not greater than tn.…”

Analysis of a combined barycentric finite volume—nonconforming finite element method for nonlinear convection-diffusion problems

“…The upwind version of the Crouzeix-Raviart finite element method was developed and analyzed in [27] for a linear stationary convection-diffusion equation. This was the inspiration for Schieweck and Tobiska who investigated in [29] upwind schemes for the steady incompressible Navier-Stokes equations.…”

“…There is an extensive literature on the numerical solution of convection-diffusion problems. Let us mention, e.g., the papers [1], [2], [22], [23], [27], [29], [32], [34], [35], the monographs [26], [28] and the references therein, devoted mainly to linear problems. The main difficulty which must be overcome is the accurate resolution of the so-called boundary layers.…”