An ILU preconditioner with coupled node fill-in for iterative solution of the mixed finite element formulation of the 2D and 3D Navier-Stokes equations

In the present paper, preconditioning of iterative equation solvers for the Navier-Stokes equations is investigated. The Navier-Stokes equations are solved for the mixed finite element formulation. The linear equation solvers used are the orthomin and the Bi-CGSTAB algorithms. The storage structure of the equation matrix is given special attention in order to avoid swapping and thereby increase the speed of the preconditioner. The preconditioners considered are Jacobian, SSOR and incomplete LU preconditioning of the matrix associated with the velocities. A new incomplete LU preconditioning with fill-in for the pressure matrix at locations in the matrix where the corner nodes are coupled is designed. For all preconditioners, inner iterations are investigated for possible improvement of the preconditioning. Numerical experiments are executed both in two and three dimensions.     

A transient solution method for the finite element incompressible Navier-Stokes equations

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a direct time integration method which has unique features in its formulation as well as in its evaluation of the contribution of external functions. Particular processes regarding the continuity conditions and the boundary conditions lead to a set of non-linear recurrence equations which represent evolution of the velocities and the pressures under the incompressibility constraint. An iteration process as to the non-linear convective terms is performed until the convergence is achieved in every integration step. Excessively artificial techniques are not introduced into the present solution procedure. Numerical examples with vortex shedding behind a rectangular cylinder are presented to illustrate the features of the proposed method. The calculated results are compared with experimental data and visualized flow fields in literature.     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.     

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method (FEM) for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q ₁ ^rot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H ¹-norm and the pressure in the L ²-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

A finite element method to solve the Navier-Stokes equations using the method of characteristics

We present a simple and efficient finite element method to solve the Navier-Stokes equations in primitive variables V, p. It uses (a) an explicit advection step, by upwind differencing. Improvement with regard to the classical upwind differencing scheme of the first order is realized by accurate calculation of the characteristic curve across several elements, and higher order interpolation; (b) an implicit diffusion step, avoiding any theoretical limitation on the time increment, and (c) determination of the pressure field by solving the Poisson equation. Two laminar flow calculations are presented and compared to available numerical and experimental results.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations

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Discrete formulation of mixed finite element methods for vapor deposition chemical reaction equations

The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element (MFE) method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence (error estimate) of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.     

A decomposed Crouzeix-Raviart element for the finite element solution of the Navier-Stokes equation

A methodology for the decomposition of the Crouzeix-Raviart finite element into six linear subelements is described. The resulting element is shown to satisfy the Brezzi-Babu?ka compatibility condition. The error bounds are also established. A comparison in accuracy between this and the standard Crouzeix-Raviart element is presented for driven cavity flows. Other results include the execution time for the DCR element and the Crouzeix-Raviart element along with both analytical and numerical integration. It is shown that the decomposed element results in shorter execution times with only marginal changes in accuracy.     

A finite element solutionof three-dimensional mixed convection gas flows in horizontal chnnels using preconditioned iterative metrix methods

An algorithm is presented for the finite element solution of three-dimensional mixed convection gas flows in channels heated from below. The algorithm uses Newton's method and iterative matrix methods. Two iterative solution algorithms, conjugate gradient squared (CGS) and generalized minimal residual (GMERS), are used in conjunction with a preconditioning technique that is simple to implement. The preconditioner is a subset of the full Jacobian matrix centered around the main diagonal but retaining the most fundamental axial coupling of the residual equations. A domain-renumbering scheme that enhances the overall algorithm performance is proposed on the basis of and analysis of the preconditioner. Comparison with the frontal elimination method demonstrates that the iterative method will be faster when the front width exceeds approximately 500. Techniques for the direct assembly f the problem into a compressed sparse row storage format are demonstrated. Elimination of fixed boundary conditions is shown to decrease the size of the matrix problem by up to 30%. Finally, fluid flow solutions obtained with the numerical technique are presented. These solutions reveal complex three-dimensional mixed convection fluid flow phenomena at low Reynolds numbers, including the reversal of the direction of longitudinal rolls in the presence of a strong recirculation in the entrance region of the channel.     

Numerical solution of the Navier-Stokes equations using a finite element method

A finite element algorithm for solving the Navier-Stokes equations is presented for the analysis of high-speed viscous flows. The algorithm uses triangular elements. The unsteady equations are integrated to steady state with a Runge-Kutta time-marching scheme. A postprocessing artificial dissipation term is introduced to stabilize the computations and to dampen dissipation errors. Numerical results are compared with the calculation of uniform flow on a rectangular region which encounters an embedded oblique shock. A shock/turbulent boundary layer problem is also solved and results are compared with experimental data. It is shown that the postprocessing smoothing term and boundary conditions similar to the finite difference method work well in the present numerical studies.     

A finite element method for the three-dimensional non-steady Navier-Stokes equations

The algorithm for solving the three-dimensional non-steady Navier-Stokes equations by the explicit forward Euler method is shown and the Galerkin finite element formulation is presented. As a numerical example, an entrace flow in a square duct is illustrated.     

A modified finite element method for solving the time-dependent,incompressible Navier-Stokes equations. Part 1: Theory

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection-dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid-driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.     

Higher-order streamfunction-vorticity formulation of 2D steady-state Navier-Stokes equations

Based on regular boundary element method and a kind of linearity invariance under homotopy, a kind of numerical scheme of 2D steady-state Navier-Stokes equation in streamfunction-vorticity formulation is described. The flow inside a square cavity is used to illustrate this numerical scheme.     

A review of some finite element methods to solve the stationary Navier-Stokes equations

In this paper the integrated solution approach, the penalty function approach and the solenoidal approach for the finite element solution of the stationary Navier-Stokes equations are compared. It is shown that both the penalty function approach and the solenoidal approach compare favourably to the integrated solution method. For fine meshes the solenoidal approach appears to be the cheapest method.     

A deterministic vortex method for solving the Navier-Stokes equations

In this paper, a new 2-D vortex method is developed, which treats the vorticity diffusion in a deterministical way. The Laplacian operator, which describes vorticity diffusion, is approximated by a contour integral. The numerical results of two model problems show that this method has a good accuracy. A primary error estimation is given, and the self-adaptive vortex blob and the boundary conditions are discussed. The project supported by the National Natural Science Foundation of China     

Mixed finite element method of hexahedral elements for Navier-Stokes problem

In this paper, we derive a new mixed element format of hexahedral elements for Navier-Stokes problem in three-dimensional space.     

Linear and non-linear iterative methods for the incompressible Navier-Stokes equations

In this study, the discretized finite volume form of the two-dimensional, incompressible Navier-Stokes equations is solved using both a frozen coefficient and a full Newton non-linear iteration. The optimal method is a combination of these two techniques. The linearized equations are solved using a conjugate-gradient-like method (CGSTAB). Various types of preconditioning are developed. Completely general sparse matrix methods are used. Investigations are carried out to determine the effect of finite volume cell anisotropy on the preconditioner. Numerical results are given for several test problems.