A quadratic equal‐order stabilized method for Stokes problem based on two local Gauss integrations

Abstract

In this article, we analyze a quadratic equal-order stabilized finite element approximation for the incompressible Stokes equations based on two local Gauss integrations. Our method only offsets the discrete pressure gradient space by the residual of the simple and symmetry term at element level to circumvent the inf-sup condition. And this method does not require specification of a stabilization parameter, and always leads to a symmetric linear system. Furthermore, this method is unconditionally stable, and can be implemented at the element level with minimal additional cost. Finally, we give some numerical simulations to show good stability and accuracy properties of the method.

An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

“…The next theorem shows the continuity property and the weak coercivity property of the bilinear form ℎ ((u ℎ , ℎ ), (k, )) for the finite element space V ℎ × ℎ in [35] and ℎ ((u ℎ , ℎ ); (V, )) for the finite element space V ℎ × ℎ in [30,32].…”

“…It is known that this choice of the approximate spaces M 1 ℎ × ℎ or M 2 ℎ × ℎ satisfies the inf-sup condition in [38], but this choice of the approximate spaces V ℎ × ℎ or V ℎ × ℎ does not satisfy the inf-sup condition [30,32,35]. As a consequence, we give a stabilized finite element approximation based on local Gauss integration technique (see [32,35]).…”

“…As a consequence, we give a stabilized finite element approximation based on local Gauss integration technique (see [32,35]). The idea is as follows.…”

“…In order to avoid the instability problem, the stabilized finite element methods are applied to the incompressible flow. Therefore, a lot of work focuses on stabilization (see [30][31][32][33][34][35][36][37]) of the lowest equal order pairs. Particularly, based on the work of Bochev et al [30], Li et al [31,32] used the projection of the pressure onto the piecewise constant space to add the stabilized term for 1 − 1 element and Zheng et al [35] used the projection of the pressure-gradient onto the piecewise constant space to add the stabilized term for 2 − 2 element.…”

An efficient two-step algorithm for the incompressible flow problem

“…Therefore, it is gaining more and more popularity in computational fluid dynamics. Moreover, based on the work of Bochev, Li et al have focused on stabilization of the lowest equal-order finite element pair P 1 − P 1 (linear functions) using the projection of the pressure onto the piecewise constant space [25][26][27][28], of the quadratic equal-order finite element pair P 2 − P 2 (quadratic functions) using the projection of the pressure-gradient onto the piecewise constant space [35]. These methods offset the inf-sup condition by adding to the bilinear form the difference between an exact Gaussian quadrature rule for quadratic polynomials and an exact Gaussian quadrature rule for linear polynomials.…”