A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

2014 ◽  
Vol 6 (5) ◽  
pp. 615-636 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Stokes Equations ◽  
Volume Element ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Fully Discrete ◽  
Finite Volume Element ◽  
Stationary Navier Stokes Equations

AbstractA semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Anas Rachid ◽  
Mohamed Bahaj ◽  
Noureddine Ayoub
Keyword(s):  
Finite Volume ◽  
Stokes Equations ◽  
Finite Volume Methods ◽  
Volume Element ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Fine Mesh ◽  
Finite Volume Element ◽  
Volume Approximation

We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths.

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