Existence of a weak solution to the Navier–Stokes equation in a general time‐varying domain by the Rothe method

We assume that Ω^t is a domain in ?³, arbitrarily (but continuously) varying for 0?t?T. We impose no conditions on smoothness or shape of Ω^t. We prove the global in time existence of a weak solution of the Navier–Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q_{[0, T)} := {( x , t);0?t?T, x ∈Ω^t}. The solution satisfies the energy‐type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright © 2008 John Wiley & Sons, Ltd.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐perturbative solution of three‐dimensional Navier–Stokes equations for the flow near an infinite rotating disk

In this paper, we present Homotopy perturbation method (HPM) and Padé technique, for finding non‐perturbative solution of three‐dimensional viscous flow near an infinite rotating disk. We compared our solution with the numerical solution (fourth‐order Runge–Kutta). The results show that the HPM–Padé technique is an appropriate method in solving the systems of nonlinear equations. The mathematical technique employed in this paper is significant in studying some other problems of engineering. Copyright © 2009 John Wiley & Sons, Ltd.     

Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces

This note studies the well‐posedness of the fractional Navier–Stokes equations in some supercritical Besov spaces as well as in the largest critical spaces for β ∈ (1/2,1). Meanwhile, the well‐posedness for fractional magnetohydrodynamics equations in these Besov spaces is also studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Error analysis of a fully discrete finite element variational multiscale method for time‐dependent incompressible Navier–Stokes equations

A finite element variational multiscale method based on two local Gauss integrations is applied to solve numerically the time‐dependent incompressible Navier–Stokes equations. A significant feature of the method is that the definition of the stabilization term is derived via two local Guass integrations at element level, making it more efficient than the usual projection‐based variational multiscale methods. It is computationally cheap and gives an accurate approximation to the quantities sought. Based on backward Euler and Crank–Nicolson schemes for temporal discretization, we derive error bounds of the fully discrete solution which are first and second order in time, respectively. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A two-grid method based on Newton iteration for the Navier–Stokes equations

In this paper, we consider a two-grid method for resolving the nonlinearity in finite element approximations of the equilibrium Navier–Stokes equations. We prove the convergence rate of the approximation obtained by this method. The two-grid method involves solving one small, nonlinear coarse mesh system and two linear problems on the fine mesh which have the same stiffness matrix with only different right-hand side. The algorithm we study produces an approximate solution with the optimal asymptotic in h and accuracy for any Reynolds number. Numerical example is given to show the convergence of the method.     

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.     

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

Parabolic Dirac operators and the Navier–Stokes equations over time‐varying domains

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in‐stationary Navier–Stokes equations over time‐varying domains. Copyright © 2005 John Wiley & Sons, Ltd.     

Drag minimization for Navier–Stokes Flow

This paper investigates the drag minimization in a two‐dimensional flow which is governed by a nonhomogeneous Navier–Stokes equations. Two approaches are utilized to derive shape gradient of the cost functional. The first one is to use the shape derivative of the fluid state and its associated adjoint state; the second one is to utilize the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally, a gradient type algorithm is effectively formulated and implemented for the mentioned drag minimization problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Moment estimates for weak solutions to the Navier–Stokes equations in half‐space

We establish the moment estimates for a class of global weak solutions to the Navier–Stokes equations in the half‐space. Copyright © 2009 John Wiley & Sons, Ltd.     

Variational iteration method for solving three‐dimensional Navier–Stokes equations of flow between two stretchable disks

The similarity transform for the steady three‐dimensional Navier–Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations. In this article, the variational iteration method was used for solving these equations. The results have been compared with the numerical results. This article depicts that the VIM is an efficient and powerful method for solving nonlinear differential equations. This method is applicable to strongly and weakly nonlinear problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

The aim of the present work is to find the numerical solutions for time‐fractional coupled Burgers equations using a new novel technique, called fractional natural decomposition method (FNDM). Two examples are considered in order to illustrate and validate the efficiency of the proposed algorithm. The numerical simulation has been conducted to ensure the exactness of the present method, and the obtained solutions are offered graphically to reveal the applicability and reliability of the FNDM. The outcomes of the study reveal that the FNDM is computationally very effective and accurate to study the (2 + 1)‐dimensional coupled Burger equations of arbitrary order.     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

Variational iteration method for solving the space‐ and time‐fractional KdV equation

This paper presents numerical solutions for the space‐ and time‐fractional Korteweg–de Vries equation (KdV for short) using the variational iteration method. The space‐ and time‐fractional derivatives are described in the Caputo sense. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via variational theory. The iteration method, which produces the solutions in terms of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and accurate when applied to space‐ and time‐fractional KdV equations. The method introduces a promising tool for solving many space–time fractional partial differential equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

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.     

Local projection FEM stabilization for the time‐dependent incompressible Navier–Stokes problem

We consider conforming finite element (FE) approximations of the time‐dependent, incompressible Navier–Stokes problem with inf‐sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence‐free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39–77 for edge‐stabilized FE approximation of the Navier–Stokes problem. Some numerical experiments complement the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1224–1250, 2015     

A linear,decoupled fractional time‐stepping method for the nonlinear fluid–fluid interaction

In this paper, a linear decoupled fractional time stepping method is proposed and developed for the nonlinear fluid–fluid interaction governed by the two Navier–Stokes equations. Partitioned time stepping method is applied to two‐physics problems with stiffness of the coupling terms being treated explicitly and is also unconditionally stable. As for each fluid, the velocity and pressure are respectively determined by just solving one vector‐valued quasi‐elliptic equation and the Possion equation with homogeneous Neumann boundary condition per time step. Therefore, the cost of the fluid–fluid interaction is dominant to solve four simple linear equations, which greatly reduces the computational cost of the whole system. The method exploits properties of the fluid–fluid system to establish its stability and convergence with the same results as the standard scheme. Finally, numerical experiments are presented to show the performance of the proposed method.     

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

We present a second‐order ensemble method based on a blended three‐step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second‐order ensemble method that combines the two‐step BDF timestepping scheme and a special explicit second‐order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 34–61, 2017     

A fully discrete nonlinear Galerkin method for the 3D Navier–Stokes equations

The purpose of this paper is twofold: (i) We show that the Fourier‐based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier–Stokes equations in three space dimensions provided the large scale/small scale cutoff is appropriately chosen. (ii) If smoothness is assumed, NLGM always outperforms the Galerkin method by a factor equal to 1 in the convergence order of the H ¹‐norm for the velocity and the L²‐norm for the pressure. This is a purely linear superconvergence effect resulting from standard elliptic regularity and holds independently of the nature of the boundary conditions (whether periodicity or no‐slip BC is enforced). © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008