Abstract Besov space approach to the non-stationary Navier–Stokes equations

The local existence, uniqueness and regularity of solutions of the initial-value problem for non-stationary Navier–Stokes equations are studied via abstract Besov spaces. The assumptions about Ω the initial data and the external force are improved on by rather simple proofs.     

A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations

The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q₁ (CNRQ₁) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H¹‐norm and the pressure in L²‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016     

Iterative methods in penalty finite element discretizations for the Steady Navier–Stokes equations

Three penalty finite element methods are designed to solve numerically the steady Navier–Stokes equations, where the Stokes, Newton, and Oseen iteration methods are used, respectively. Moreover, the stability analysis and error estimate for these nine algorithms are provided. Finally, the numerical tests confirm the theoretical results of the presented algorithms. Meanwhile, the numerical investigations are provided to show that the proposed methods are efficient for solving the steady Navier–Stokes equations with the different viscosity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 74‐94, 2014     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Study of the mixed finite volume method for Stokes and Navier‐Stokes equations

We present finite volume schemes for Stokes and Navier‐Stokes equations. These schemes are based on the mixed finite volume introduced in (Droniou and Eymard, Numer Math 105 (2006), 35‐71), and can be applied to any type of grid (without “orthogonality” assumptions as for classical finite volume methods) and in any space dimension. We present numerical results on some irregular grids, and we prove, for both Stokes and Navier‐Stokes equations, the convergence of the scheme toward a solution of the continuous problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Time stepping procedures for the non-stationary Stokes equations

We consider first and second-order implicit time stepping procedures for the non-stationary Stokes equations in bounded domains of ?³. Using energy estimates we prove the optimal convergence properties in the Sobolev spaces H^m(G)(m = 0, 1, 2) uniformly in time, provided that the Stokes solution has a certain degree of regularity. Here in the case of the second-order scheme (method of Crank–Nicholson) the Stokes solution has to satisfy a non-local compatibility condition at the initial time t = O, which can be satisfied by a special initial construction.     

A nearly optimal preconditioner for the Navier–Stokes equations

We present a preconditioner for the linearized Navier–Stokes equations which is based on the combination of a fast transform approximation of an advection diffusion problem together with the recently introduced ‘BFBT^T’ preconditioner of Elman (SIAM Journal of Scientific Computing, 1999; 20 :1299–1316). The resulting preconditioner when combined with an appropriate Krylov subspace iteration method yields the solution in a number of iterations which appears to be independent of the Reynolds number provided a mesh Péclet number restriction holds, and depends only mildly on the mesh size. The preconditioner is particularly appropriate for problems involving a primary flow direction. Copyright © 2001 John Wiley & Sons, Ltd.     

Coupling method for the exterior stationary Navier–Stokes equations

We represent a new numerical method to solve the stationary Navier–Stokes equations in an unbounded domain. This technique consists in coupling the boundary integral and finite element methods. Moreover, we derive the variational formulation and well-posedness of the coupling method and provide the convergence result for the approximate solution. © 1993 John Wiley & Sons, Inc.     

Stabilized spectral element approximation for the Navier–Stokes equations

The conforming spectral element methods are applied to solve the linearized Navier–Stokes equations by the help of stabilization techniques like those applied for finite elements. The stability and convergence analysis is carried out and essential numerical results are presented demonstrating the high accuracy of the method as well as its robustness. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 115–141, 1998     

Navier–Stokes equations with slip boundary conditions

In this paper, we consider incompressible viscous fluid flows with slip boundary conditions. We first prove the existence of solutions of the unsteady Navier–Stokes equations in n‐spacial dimensions. Then, we investigate the stability, uniqueness and regularity of solutions in two and three spacial dimensions. In the compactness argument, we construct a special basis fulfilling the incompressibility exactly, which leads to an efficient and convergent spectral method. In particular, we avoid the main difficulty for ensuring the incompressibility of numerical solutions, which occurs in other numerical algorithms. We also derive the vorticity‐stream function form with exact boundary conditions, and establish some results on the existence, stability and uniqueness of its solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

A two-level finite element method for the Navier–Stokes equations based on a new projection

We consider a fully discrete two-level approximation for the time-dependent Navier–Stokes equations in two dimension based on a time-dependent projection. By defining this new projection, the iteration between large and small eddy components can be reflected by its associated space splitting. Hence, we can get a weakly coupled system of large and small eddy components. This two-level method applies the finite element method in space and Crank–Nicolson scheme in time. Moreover,the analysis and some numerical examples are shown that the proposed two-level scheme can reach the same accuracy as the classical one-level Crank–Nicolson method with a very fine mesh size h by choosing a proper coarse mesh size H. However, the two-level method will involve much less work.     

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.     

Two-level Galerkin–Lagrange multipliers method for the stationary Navier–Stokes equations

In this paper, we combine the Galerkin–Lagrange multiplier (GLM) method with the two-level method to solve the stationary Navier–Stokes equations in order to avoid the time-consuming process and the construction of zero-divergence elements. Different quadrilateral partitions are used for approximating the velocity and the pressure. Then some error estimates are obtained and some numerical results of the GLM method and the two-level GLM method are given. The results show that the two-level method based on the GLM method is more efficient than the GLM method under the convergence rate of same order.     

Logarithmically improved regularity criteria for Navier–Stokes and related equations

We use an interpolation inequality on Besov spaces to show some logarithmically improved regularity criteria for Navier–Stokes equations, the harmonic heat flow, the Landau–Lifshitz equations, and the Landau–Lifshitz–Maxwell system. Copyright © 2009 John Wiley & Sons, Ltd.     

A note on integration of the Navier–Stokes equations

In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet.     

Some global regular solutions to Navier–Stokes equations

First the existence of global regular two‐dimensional solutions to Navier–Stokes equations in a bounded cylinder and for boundary slip conditions is proved. Next stability of sum of two dimensional and axially symmetric solutions is proved. Copyright © 2006 John Wiley & Sons, Ltd.     

A posteriori error estimation and error control for finite element approximations of the time-dependent Navier–Stokes equations

We present an approach to estimate numerical errors in finite element approximations of the time-dependent Navier–Stokes equations along with a strategy to control these errors. The error estimators and the error control procedure are based on the residuals of the Navier–Stokes equations, which are shown to be comparable to error components in the velocity variable. The present methodology applies to the estimation of numerical errors due to the spatial discretization only. Its performance is demonstrated for two-dimensional channel flows past a cylinder in the periodic regime.     

Nonstationary flow for the Navier–Stokes equations in a cylindrical pipe

In cylindrical domain, we consider the nonstationary flow with prescribed inflow and outflow, modelled with Navier–Stokes equations under the slip boundary conditions. Using smallness of some derivatives of inflow function, external force and initial velocity of the flow, but with no smallness restrictions on the inflow, initial velocity neither force, we prove existence of solutions in . Copyright © 2012 John Wiley & Sons, Ltd.     

A posteriori error estimators for the steady incompressible Navier–Stokes equations

A residual-based a posteriori error estimator for finite element discretizations of the steady incompressible Navier–Stokes equations in the primitive variable formulation is discussed. Though the estimator is similar to existing ones, an alternate derivation is presented, involving an abstract estimate that may prove of some intrinsic value. The estimator is particularized to Hood–Taylor and modified Hood–Taylor finite elements and proved to be a global upper bound (up to a positive multiplicative constant) of the true error. Numerical examples are provided to illustrate the performance of the resulting mesh adaptation process. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 561–574, 1997     

Strong solutions for the nonhomogeneous Navier–Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.