A finite element variational multiscale method for incompressible flows based on two local gauss integrations

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

“…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.…”

A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

“…Here, we consider a recent finite element variational multiscale method based on two local Gauss integrations [33]. In this method, a stabilization term defined by the difference between a consistent and an underintegrated matrix involving the velocity gradient is used to stabilize the numerical form of the Navier-Stokes equations.…”

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

“…There are many literatures on stabilized finite element methods for NavierStokes equations. Among them, we list some methods as follows: recently developed stabilized methods, such as, Galerkin least square method introduced in [4][5][6] by Franca, Hughes, and coworkers, and applied to some advective-diffusive models; residual-free bubbles (RFB) method [7][8][9], in which, the enrichment of the discrete finite element space by RFB; classical large eddy simulation (LES) approach in [10,11] which treats the large scales as an average in space given by convolution with an appropriate filter function; variational multiscale (VMS) methods, see for example, Hughes et al [12][13][14], they first reported VMS methods, Guermond [15] developed the subgrid modeling that is a variant of VMS methods, and Layton [16] discussed the connection between subgrid scale eddy viscosity and mixed methods, John, Kaya, and coworkers [17][18][19] applied the VMS methods to the Navier-Stokes equations and gave the theoretical error analysis, Zheng [20] improved the VMS method for the Navier-Stokes equations based on two local Gauss integrations, and other literatures on VMS methods [21][22][23][24]; two-level stabilization scheme [25] and local projection stabilization [26], both can be interpreted as a VMS method; and so on.…”

“…The two local Gauss integrations method was first developed to offset the discrete pressure space by the residual of the simple and symmetry term at element level to circumvent the inf-sup condition (see, e.g., [28,29]). Based on the observation that this method is equivalent with, but save a lot of computational CPU time than the VMS method for the Taylor-Hood elements in the case of selecting lower order appropriate space for the velocity deformation tensor on the same mesh, Zheng et al provided a one-level method for the Navier-Stokes equations [20]. They also extend their work by combing with adaptive strategy [22].…”