THE ASYMPTOTIC BEHAVIOUR OF SOME SEMI-DISCRETE AND FULLY-DISCRETE FINITE ELEMENT SCHEMES FOR THE EVOLUTION NAVIER-STOKES EQUATIONS

We present enhanced physics-based finite element schemes for two families of turbulence models, the models and the Stolz-Adams approximate deconvolution models. These schemes are delicate extensions of a method created for the Navier-Stokes equations in Rebholz (2007), that achieve high physical fidelity by admitting balances of both energy and helicity that match the true physics. The schemes' development requires carefully chosen discrete curl, discrete Laplacian, and discrete filtering operators, in order to permit the necessary differential operator commutations.

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

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2013.m83 ◽

2014 ◽

Vol 6 (5) ◽

pp. 615-636 ◽

Cited By ~ 1

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.

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Fully Discrete Finite Element Approximation for the Stabilized Gauge-Uzawa Method to Solve the Boussinesq Equations

Journal of Applied Mathematics ◽

10.1155/2013/372906 ◽

2013 ◽

Vol 2013 ◽

pp. 1-21

Author(s):

Jae-Hong Pyo

Keyword(s):

Finite Element ◽

Stokes Equations ◽

Boussinesq Equations ◽

Finite Element Approximation ◽

Navier Stokes ◽

Element Approximation ◽

Uzawa Method ◽

Element Space ◽

Fully Discrete ◽

Discrete Finite Element

The stabilized Gauge-Uzawa method (SGUM), which is a 2nd-order projection type algorithm used to solve Navier-Stokes equations, has been newly constructed in the work of Pyo, 2013. In this paper, we apply the SGUM to the evolution Boussinesq equations, which model the thermal driven motion of incompressible fluids. We prove that SGUM is unconditionally stable, and we perform error estimations on the fully discrete finite element space via variational approach for the velocity, pressure, and temperature, the three physical unknowns. We conclude with numerical tests to check accuracy and physically relevant numerical simulations, the Bénard convection problem and the thermal driven cavity flow.

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