Discontinuous Galerkin finite element approximation of the two-dimensional Navier-Stokes equations in stream-function formulation

2006 ◽  
Vol 23 (6) ◽  
pp. 447-459 ◽  
Author(s):  
Igor Mozolevski ◽  
Endre Süli ◽  
Paulo Rafael Bösing
Keyword(s):  
Stokes Equations ◽  
Finite Element Approximation ◽  
Navier Stokes ◽  
Element Approximation ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Function Formulation ◽  
Stream Function Formulation

scholarly journals Stochastic 2D Incompressible Navier-Stokes Solver Using the Vorticity-Stream Function Formulation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Mohamed A. El-Beltagy ◽  
Mohamed I. Wafa
Keyword(s):  
Stream Function ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Function Formulation ◽  
Mean And Variance ◽  
The Mean ◽  
Stream Function Formulation

A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-centered finite-volume solver. A second-order upwind scheme is used in the convection term for numerical stability and higher-order discretization. The resulting sparse linear system is solved efficiently by a direct parallel solver. The mean and variance simulations of the cavity flow are done for random variation of the viscosity and the lid velocity. The solver was tested and compared with the Monte-Carlo simulations and with previous research works. The developed solver is proved to be efficient in simulating the stochastic two-dimensional incompressible flows.

Two-Level Spline Approximations for Two-Dimensional Navier–Stokes Equations

2016 ◽  
Vol 16 (3) ◽  
pp. 497-506
Author(s):  
Xinping Shao ◽  
Danfu Han ◽  
Xianliang Hu
Keyword(s):  
Stokes Equations ◽  
Newton Iteration ◽  
Spline Space ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Function Formulation ◽  
The Matrix ◽  
Order Spline ◽  
Stream Function Formulation

AbstractThe $C^{1}$ spline spaces with degree $d\geq 5$ over given triangulations are implemented in the framework of multi-variate spline theory. Based on this approach, two-level methods are proposed by using various order spline spaces for the steady state Navier–Stokes equations in the stream function formulation. The proposed method can be reduced to solving a linear equation in the high-order spline space and the nonlinear equations in the low-order spline space. The convergence analysis is given based on the Newton iteration. Besides, the matrix forms of the two-level scheme are also presented. We finally tabulate the numerical results to validate and show the efficiency of the proposed two-level spline methods.

FINITE ELEMENT ERROR ANALYSIS OF SPACE AVERAGED FLOW FIELDS DEFINED BY A DIFFERENTIAL FILTER

2004 ◽  
Vol 14 (04) ◽  
pp. 603-618 ◽  
Author(s):  
ADRIAN DUNCA ◽  
VOLKER JOHN
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Flow Fields ◽  
Navier Stokes ◽  
Element Approximation ◽  
Navier Stokes Equations ◽  
Finite Element Approximations ◽  
Three Space ◽  
Element Error

This paper analyzes finite element approximations of space averaged flow fields which are given by filtering, i.e. averaging in space, the solution of the steady state Stokes and Navier–Stokes equations with a differential filter. It is shown that [Formula: see text], the error of the filtered velocity [Formula: see text] and the filtered finite element approximation of the velocity [Formula: see text], converges under certain conditions of higher order than [Formula: see text], the error of the velocity and its finite element approximation. It is also proved that this statement stays true if the L2-error of finite element approximations of [Formula: see text] and [Formula: see text] is considered. Numerical tests in two and three space dimensions support the analytical results.

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