Penalty finite element method for the Navier-Stokes equations

doi:10.1016/0045-7825(84)90025-2

Computer Methods in Applied Mechanics and Engineering

Volume 42, Issue 2, February 1984, Pages 183-224

https://doi.org/10.1016/0045-7825(84)90025-2 Get rights and content

Abstract

We present an analysis of a penalty formulation of the stationary Navier-Stokes equations for an incompressible fluid. Subject to restrictions on the viscosity and prescribed body force, it is shown that there exists a unique solution to this penalty problem. The solution to the penalty problem is shown to converge to the solution of the Navier-Stokes problem as O(ε) where ε → 0 is the penalty parameter.

Existence, uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized Navier-Stokes problem presented here. Numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases.

References (19)

M. Bercovier et al.
A finite element method for the numerical solution of viscous incompressible flows
J. Comput. Phys.
(1979)
G.F. Carey et al.
Penalty approximation of Stokes flow
Comput. Meths. Appl. Mech. Engrg.
(1982)
T.J. Hughes et al.
Finite element analysis of incompressible viscous flows by the penalty function formulation
J. Comput. Phys.
(1979)
F. Brezzi
On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers
RAIRO Numer. Anal.
(1974)
G.F. Carey et al.
Convergence of iterative methods in penalty finite element approximation of Navier-Stokes equations
Comput. Meths. Appl. Mech. Engrg.
(1984)
V. Girault et al.
Finite Element Approximation of the Navier-Stokes Equations
O.A. Ladyzhenskaya
The Mathematical Theory of Viscous Incompressible Flow
(1964)
C.B. Morrey
Multiple Integrals in the Calculus of Variations
(1966)
J.T. Oden et al.
Finite Elements: Mathematical Aspects
(1982)

There are more references available in the full text version of this article.

Cited by (46)

Flow stability and regime transitions on periodic open foams
2024, International Journal of Multiphase Flow
This study aims to link critical Reynolds numbers associated with either steady-state or temporal bifurcations to fluid flow regimes described by macroscopic laws (Darcy, Forchheimer) in a periodic 3D Kelvin foam, using direct numerical simulations. We first identify the permeability and inertial coefficient of the Darcy’s law and the Forchheimer’s one. We explicit the different flow regimes that are accounted for in the Forchheimer’s framework (Darcian, weak inertia and strong inertia regimes, respectively). We present an original systematic way to determine the critical Reynolds number associated with both regime transitions. We calculate them over a wide range of porosities and present two power-law correlations that locate these regime transitions for engineering purposes.
We have performed pore-scale resolved calculations for various porosities with the Asymptotic Numerical Method (ANM) to find steady-state bifurcations, if any, along with a Linear Stability Analysis (LSA) to find temporal (Hopf) bifurcations from steady-state base flows. All the computed bifurcations occur at Reynolds numbers in the vicinity of the transition from weak to strong inertia regimes, where a change of behavior takes place. On the other hand, no bifurcation has been found in the transition between Darcian and weak inertia regimes.
A decoupling penalty finite element method for the stationary incompressible MagnetoHydroDynamics equation
2019, International Journal of Heat and Mass Transfer
In this paper, we give a penalty finite element method for the steady MHD equations. In this method, we decouple the MHD into two equations, one for the velocity and magnetic $(u, B)$ , the other for the pressure p. We prove the existence of the penalty method and the optimal error estimate. Then, we give the penalty finite element method for the MHD equations. The stability analysis shows our method is stable, and the error estimate shows our method has an optimal convergence order. Finally, we give some numerical results of exact solution equation and Hartmann flow. The numerical results show that our penalty finite element method is effect.
An error analysis of the finite element method overcoming corner singularities for the stationary Stokes problem
2017, Computers and Mathematics with Applications
In this paper, we introduce a mixed finite element method to overcome corner singularities of the stationary Stokes system on a non-convex polygon. The reference Choi and Kweon (2013) says that the velocity field and the pressure function are composed of singular parts and regular remainders near a non-convex vertex, and furthermore the regular parts are sufficiently smooth on the polygon. We use the corner singularity expansion and propose new extraction formulae for coefficients of singularities. For the proposed numerical method, we first try to find finite element solutions for the regular parts and then calculate approximations of the coefficients by the derived formulae expressed by the remainders and given functions. We show error estimates of the approximations for the regular parts and coefficients, and give some numerical experiments to confirm the efficiency and reliability of the proposed method.
A finite element method for singular solutions of the Navier-Stokes equations on a non-convex polygon
2016, Journal of Computational and Applied Mathematics
Citation Excerpt :
Also, error estimates and iterative convergence results for the nonlinear equations were shown. In [16] a penalty formulation for the Navier–Stokes equations was analyzed, and in [17] a nonconforming finite element method for the inhomogeneous boundary value problem was constructed. In [18] a stabilization technique allowing the use of equal interpolation for the velocity and pressure was analyzed, and in [19] the mixed formulation of the Navier–Stokes equations with mixed boundary conditions in 2D polygonal domains was constructed.
It is shown in Choi and Kweon (2013) that a solution of the Navier–Stokes equations with no-slip boundary condition on a non-convex polygon can be written as $[u, p] = C_{1} [Φ_{1}, ϕ_{1}] + C_{2} [Φ_{2}, ϕ_{2}] + [u_{R}, p_{R}]$ near each non-convex vertex, where $[u_{R}, p_{R}] \in H^{2} \times H^{1}$ , $[Φ_{i}, ϕ_{i}]$ are corner singularity functions for the Stokes problem with no-slip condition, and $C_{i} \in R$ are coefficients which are called the stress intensity factors. We design a finite element method to approximate the coefficients $C_{i}$ and the regular part $[u_{R}, p_{R}]$ , show the unique existence of the approximations, and derive their error estimates. Some numerical examples are given, confirming convergence rates for the approximations.
Dynamic interaction of heat transfer, air flow and disc vibration of disc drives - Theoretical development and numerical analysis
2014, International Journal of Mechanical Sciences
Citation Excerpt :
In the study of the penalty finite element method for Navier–Stokes equations, it was demonstrated systematically by Codina [46], Carey and Krishnan [47,48] that “In computations with smaller values of ε, roundoff effects due to computation in finite precision are found to be an important consideration” and “Perhaps the only drawback of this approach (of the penalty finite element method) is the ill-conditioning of the stiffness matrix when the penalty parameter ε is very small”. Just theoretically speaking, the solution from the penalty finite element method can converge to its true value when penalty parameter ε goes to zero, whereas the fact is that “These theoretical results presume exact arithmetic, not limited by the finite precision of digital computation” as stated in Carey and Krishnan [47]. As a result, when penalty parameter ε is very small, starting from 10−7 in Table 5, “the roundoff effects” and “the ill-conditioning of the stiffness matrix” emerge and cause the ‘stiffness’ matrix to become close to singular or badly scaled and some inaccurate results are obtained.
Steady state air flow, heat transfer and thermoelastic dynamics in the multi-field coupling system of a rotating flexible disc in an enclosure filled with air are investigated within a large speed range. The system represents rotating discs in hard disc drives and optical disc drives. With Navier–Stokes and continuity equations and an improved penalty finite element method, air velocities and pressure induced by disc rotation in the enclosure are obtained. Temperature distribution of the rotating disc, driving shaft, enclosure and air flow is determined under the external heat sources from the shaft-driving motor and the circuit board inside the enclosure, the internal heat source from aerodynamic heating due to viscous dissipation of fluid, the heat convection in air flow, and the free convection heat loss at the enclosure׳s outside surfaces. Natural frequencies of the rotating disc are solved under the stresses induced by the disc temperature distribution and centrifugal force. Effects of heat convection and aerodynamic heating induced by disc rotation on system heat balance and dynamic characteristics of the rotating disc are investigated. The method presented is helpful to design of hard/optical disc drives and many other applications that involve rotating discs in fluids and with heat transfer.
Numerical study on permeate flux enhancement by spacers in a crossflow reverse osmosis channel
2006, Journal of Membrane Science
The impact of spacer configurations (i.e. cavity, zigzag and submerged) and mesh length on the alleviation of concentration polarization and the enhancement of permeate flux in the crossflow reverse osmosis membrane channels was investigated. In this study, the wall concentration and permeate flux were directly determined from the numerical solutions of the fully coupled governing equations of momentum and mass transfer in the feed channel. It was demonstrated that the average permeate flux could be significantly enhanced by the spacers, especially those with zigzag configuration. Simulations showed that the zigzag configuration was the most effective one to alleviate concentration polarization and to enhance permeate flux while the submerged configuration is the least. It was further found that an optimum mesh length (corresponding to the maximum permeate flux enhancement) existed for cavity and zigzag configurations and the optimum mesh length decreased with increasing salinity of the feed water. The results suggested that different mesh length should be used in membrane modules for feed waters of different salinities to obtain the maximum permeate flux enhancement.

View all citing articles on Scopus

View full text

Penalty finite element method for the Navier-Stokes equations

Abstract

J. Comput. Phys.

Comput. Meths. Appl. Mech. Engrg.

J. Comput. Phys.

On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers

RAIRO Numer. Anal.

Convergence of iterative methods in penalty finite element approximation of Navier-Stokes equations

Comput. Meths. Appl. Mech. Engrg.

Finite Element Approximation of the Navier-Stokes Equations

The Mathematical Theory of Viscous Incompressible Flow

Multiple Integrals in the Calculus of Variations

Finite Elements: Mathematical Aspects