Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

Abstract

This article focuses on discontinuous Galerkin method for the two-or three-dimensional stationary incompressible Navier-Stokes equations. The velocity field is approximated by discontinuous locally solenoidal finite element, and the pressure is approximated by the standard conforming finite element. Then, superconvergence of nonconforming finite element approximations is applied by using least-squares surface fitting for the stationary Navier-Stokes equations. The method ameliorates the two noticeable disadvantages about the given finite element pair. Finally, the superconvergence result is provided under some regular assumptions.

A local parallel superconvergence method for the incompressible flow by coarsening projection

“…In this article, we pay more attention to superconvergence by coarsening projection proposed by Wang. More literature can be found in . Superconvergence by coarsening projection does not care location of the superconvergence points or clusters, problem's type, and mesh quality, this approach strongly depends on the smoothness of the exact solution and a priori regularity of the underlying problem over the whole domain.…”

Superconvergence of a stabilized approximation for the Stokes eigenvalue problem by projection method

“…The projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares surface fitting. Some details of this projection method can be found in the works of Chen and Wang [4], Heimsund et al [8], Ye et al [23], [5], [6], Liu and Yan [19], Li et al [18], [14], [13], [16], [17] and Huang et al [11]. Li et al have used a local coarse mesh L 2 -projection to establish the superconvergence of a stabilized finite element approximation for the Stokes equations.…”

Penalty finite element approximations for the Stokes equations by L2 projection

“…Furthermore, the method helps to extend these results to Navier-Stokes equations with discontinuous Galerkin method [9] or local smooth domain [7] for solving the Navier-Stokes equations that will be discussed in our further papers.…”