Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two-dimensional stationary Navier-Stokes equations is analyzed. For the P1–P0 element, we obtain the optimal L2 error estimates of the finite volume solution uh and ph. We also provide some numerical examples to confirm the efficiency of the FVM. Furthermore, the effect of initial value for iterative method is analyzed carefully.

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Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.11-m11178 ◽

2013 ◽

Vol 5 (1) ◽

pp. 19-35 ◽

Cited By ~ 3

Author(s):

Tong Zhang ◽

Shunwei Xu

Keyword(s):

Finite Volume Method ◽

Finite Volume ◽

Stokes Equations ◽

Stokes Problem ◽

Mesh Size ◽

Finite Volume Methods ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Fine Mesh ◽

Volume Method

AbstractIn this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size H, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size . These methods we studied provide an approximate solution with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.

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Collocated Finite-Volume Method for the Incompressible Navier-Stokes Problem

Journal of Numerical Mathematics ◽

10.1515/jnma-2020-0008 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Kirill M. Terekhov

Keyword(s):

Finite Volume Method ◽

Finite Volume ◽

Stokes Problem ◽

Piecewise Linear ◽

Navier Stokes ◽

Piecewise Constant ◽

Artificial Boundary Conditions ◽

Continuity Equations ◽

Volume Method ◽

Convection Dominated

AbstractThe article introduces a collocated finite-volume method for the incompressible Navier-Stokes problem. The method applies to general polyhedral grids and demonstrates higher than the first order of convergence. The velocity components and the pressure are approximated by piecewise-linear continuous and piecewise-constant fields, respectively. The method does not require artificial boundary conditions for pressure but requires stabilization term to suppress the error introduced by piecewise-constant pressure for convection-dominated problems. Both the momentum and continuity equations are approximated in a flux-conservative fashion, i.e. the conservation for both quantities is discretely exact. The attractive side of the method is a simple flux-based finite-volume construction of the scheme. Applicability of the method is demonstrated on several numerical tests using general polyhedral grids.

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