Stability of a colocated finite volume scheme for the Navier‐Stokes equations

The aim of this article is to describe a colocated finite volume approximation of the incompressible Navier‐Stokes equation and study its stability. One of the advantages of colocated finite volume space discretizations over staggered space discretizations is that all the variables share the same location; hence, the possibility to more easily use complex geometries and hierarchical decompositions of the unknowns. The time discretization used in the scheme studied here is a projection method. First, we give the full discretization of the incompressible Navier‐Stokes equations, then, we state the stability result and prove it following the methods of Marion and Temam. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

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

A combination method of two-grid discretization approach with a recent finite element variational multiscale algorithm for simulation of the incompressible Navier–Stokes equations is proposed and analyzed. The method consists of a global small-scale nonlinear Navier–Stokes problem on a coarse grid and local linearized residual problems in overlapped fine grid subdomains, where the numerical form of the Navier–Stokes equations on the coarse grid is stabilized by a stabilization term based on two local Gauss integrations at element level and defined by the difference between a consistent and an under-integrated matrix involving the gradient of velocity. By the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method.     

First‐order system least squares for the Oseen equations

Following earlier work for Stokes equations, a least squares functional is developed for two‐ and three‐dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier–Stokes equations. An algorithm is presented for approximately solving steady‐state, incompressible Navier–Stokes equations with a nested iteration‐Newton‐FOSLS‐AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X_H and X_h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ? H, respectively, and a finite element space M_h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H⁵. If we choose H = O(h^2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.     

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017     

Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate

For strong solutions of the incompressible Navier‐Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf‐sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier‐Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.     

Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations

A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Analysis of algebraic multigrid parameters for two-dimensional steady-state heat diffusion equations

In this work, it is provided a comparison for the algebraic multigrid (AMG) and the geometric multigrid (GMG) parameters, for Laplace and Poisson two-dimensional equations in square and triangular grids. The analyzed parameters are the number of: inner iterations in the solver, grids and unknowns. For the AMG, the effects of the grid reduction factor and the strong dependence factor in the coarse grid on the necessary CPU time are studied. For square grids the finite difference method is used, and for the triangular grids, the finite volume one. The results are obtained with the use of an adapted AMG1R6 code of Ruge and Stüben. For the AMG the following components are used: standard coarsening, standard interpolation, correction scheme (CS), lexicographic Gauss–Seidel and V-cycle. Comparative studies among the CPU time of the GMG, AMG and singlegrid are made. It was verified that: (1) the optimum inner iterations is independent of the multigrid, however it is dependent on the grid; (2) the optimum number of grids is the maximum number; (3) AMG was shown to be sensitive to both the variation of the grid reduction factor and the strong dependence factor in the coarse grid; (4) in square grids, the GMG CPU time is 20% of the AMG one.     

A two-grid method with backtracking for the mixed Navier–Stokes/Darcy model

In this paper, we consider the effect of adding a coarse mesh correction to the two-grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L² and H¹ optimal velocity and piezometric head approximations and an L² optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method.     

Solving the driven flow by linearized elimination with differential quadrature

In the Navier‐Stokes equations the velocity and the pressure are coupled together by the incompressibility condition div u = 0( u = (u,v)^T) which makes the equations difficult to solve numerically. In this article, a new method named linearized elimination of unknowns with differential quadrature method is applied to the Navier‐Stokes equations. The method is of second‐order accuracy in time and of spectral accuracy in space. The numerical results show that our method is of high accuracy, of good convergence with little computational efforts. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017     

Leading term at infinity of steady Navier‐Stokes flow around a rotating obstacle

Consider a viscous incompressible flow around a body in $\mathbb R^3$ rotating with constant angular velocity ω. Using a coordinate system attached to the body, the problem is reduced to a modified Navier‐Stokes system in a fixed exterior domain. This paper addresses the question of the asymptotic behavior of stationary solutions to the new system as |x| → ∞. Under a suitable smallness assumption on the velocity field, u, and the net force on the boundary, N, we prove that the leading term of u is the so‐called Landau solution U, a singular solution of the stationary Navier‐Stokes system in $\mathbb R^3$ with external force kωδ₀ and decaying as 1/|x|; here $k\in \mathbb R$ is a suitable constant determined by N and δ₀ is the Dirac measure supported in the origin.     

Parallel coarse‐grid selection

Algebraic multigrid (AMG) is a powerful linear solver with attractive parallel properties. A parallel AMG method depends on efficient, parallel implementations of the coarse‐grid selection algorithms and the restriction and prolongation operator construction algorithms. In the effort to effectively and quickly select the coarse grid, a number of parallel coarsening algorithms have been developed. This paper examines the behaviour of these algorithms in depth by studying the results of several numerical experiments. In addition, new parallel coarse‐grid selection algorithms are introduced and tested. Copyright © 2007 John Wiley & Sons, Ltd.     

Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Explicit solution of the incompressible Navier–Stokes equations with linear finite elements

A three-field finite element scheme for the explicit iterative solution of the stationary incompressible Navier–Stokes equations is studied. In linearized form the scheme is associated with a generalized time-dependent Stokes system discretized in time. The resulting system of equations allows for a stable approximation of velocity, pressure and stress deviator tensor, by means of continuous piecewise linear finite elements, in both two- and three-dimensional space. Convergence in an appropriate sense applying to this finite element discretization is demonstrated, for the stationary Stokes system.     

An algebraic multigrid method for finite element discretizations with edge elements

This paper presents an algebraic multigrid method for the efficient solution of the linear system arising from a finite element discretization of variational problems in H₀(curl,Ω). The finite element spaces are generated by Nédélec's edge elements. A coarsening technique is presented, which allows the construction of suitable coarse finite element spaces, corresponding transfer operators and appropriate smoothers. The prolongation operator is designed such that coarse grid kernel functions of the curl‐operator are mapped to fine grid kernel functions. Furthermore, coarse grid kernel functions are ‘discrete’ gradients. The smoothers proposed by Hiptmair and Arnold, Falk and Winther are directly used in the algebraic framework. Numerical studies are presented for 3D problems to show the high efficiency of the proposed technique. Copyright © 2002 John Wiley & Sons, Ltd.     

Balancing Neumann‐Neumann methods for incompressible Stokes equations

Balancing Neumann‐Neumann methods are introduced and studied for incompressible Stokes equations discretized with mixed finite or spectral elements with discontinuous pressures. After decomposing the original domain of the problem into nonoverlapping subdomains, the interior unknowns, which are the interior velocity component and all except the constant‐pressure component, of each subdomain problem are implicitly eliminated. The resulting saddle point Schur complement is solved with a Krylov space method with a balancing Neumann‐Neumann preconditioner based on the solution of a coarse Stokes problem with a few degrees of freedom per subdomain and on the solution of local Stokes problems with natural and essential boundary conditions on the subdomains. This preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively. The condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and a factor that depends on the inverse of the inf‐sup constants of the discrete problem and of the coarse subproblem. Numerical results show that the method is quite fast; they are also fully consistent with the theory. © 2002 John Wiley & Sons, Inc.