Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

On Centralizer Subalgebras of Group Algebras

Let $G$ be a finite group, $H ≤ G$ and $R$ be a commutative ring with an identity $1_R$. Let $C_{RG}(H) = \{ α ∈ RG|αh= hα$ for all $h ∈ H \}$, which is called the centralizer subalgebra of $H$ in $RG$. Obviously, if $H = G$ then $C_{RG}(H)$ is just the central subalgebra $Z(RG)$ of $RG$. In this note, we show that the set of all $H$-conjugacy class sums of $G$ forms an $R$-basis of $C_{RG}(H)$. Furthermore, let $N$ be a normal subgroup of $G$ and $γ$ the natural epimorphism from $G$ to $\overline{G}= G/N$. Then $γ$ induces an epimorphism from $RG$ to $R\overline{G}$, also denoted by $γ$. We also show that if $R$ is a field of characteristic zero, then $γ$ induces an epimorphism from $C_{RG}(H)$ to $C_{R\overline{G}}(\overline{H})$, that is, $γ(C_{RG}(H)) = C_{R\overline{G}}(\overline{H})$.     

A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.     

关系治理与契约治理导向匹配状态的量化研究——以公私合作模式下A地铁项目为例

近年来关系治理(RG)、契约治理导向(CG)是项目管理领域研究较多的议题,但是关于二者匹配状态的量化研究却很少。为此,基于问卷调研数据,利用相关性权重法对RG状态、CG状态进行了量化,在此基础上,构建了RG与CG匹配模型,并进行了案例分析。研究结果表明:对RG状态、CG状态的量化研究具有可行性,且实现了对RG与CG最佳匹配状态的量化研究。以上研究结论有助于理解RG与CG的匹配机理,对实现二者的有效匹配及项目绩效的改善具有重要的理论和实践指导意义。     

CLASSICAL QUOTIENT RINGS OF GROUP RINGS

Abstract

Throughout G will denote a free Abelian group and Z(R) the right singular ideal of a ring R. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R/Z(R) is semiprime, Z(R) is rationally closed, and Z(R) contains no closed uniform right ideals. We prove that R is a Cl-ring if and only if the group ring RG is a C1-ring. If RG has the additional property that bRG is dense whenever b is a right nonzero-divisor, then the complete ring of quotients of RG is a classical ring of quotients.     

Phase transitions in two dimensions and multiloop renormalization group expansions

We discuss using the field theory renormalization group (RG) to study the critical behavior of twodimensional (2D) models. We write the RG functions of the 2D λ? ⁴ Euclidean n-vector theory up to five-loop terms, give numerical estimates obtained from these series by Padé-Borel-Leroy resummation, and compare them with their exact counterparts known for n = 1, 0,?1. From the RG series, we then derive pseudo-ε-expansions for the Wilson fixed point location g*, critical exponents, and the universal ratio R ₆ = g ₆ /g ² , where g ₆ is the effective sextic coupling constant. We show that the obtained expansions are “friendler” than the original RG series: the higher-order coefficients of the pseudo-ε-expansions for g*, R₆, and γ ^?1 turn out to be considerably smaller than their RG analogues. This allows resumming the pseudo-ε-expansions using simple Padé approximants without the Borel-Leroy transformation. Moreover, we find that the numerical estimates obtained using the pseudo-ε-expansions for g* and γ ^?1 are closer to the known exact values than those obtained from the five-loop RG series using the Padé-Borel-Leroy resummation.     

Large time existence of small viscous surface waves without surface tension

The motion of a three-dimensional viscous, imcompressible fluid is governed by the Navier-Stokes equations. We study the case where the fluid is in an ocean of infinite extent and finite depth with a free surface on top. This gives rise to a nonlinear free boundary problem. The given data are the initial velocity field and the initial free surface. In general, given smooth data, the solution will develop singularities in finite time; however, the effect of viscosity and surface tension tends to prevent the ingulitrities. It was previously known that when both are present, small, appropriately smooth solutions do not develop singularities; that is, smooth solutions exist globally in time. In this paper, we show that viscosity alone will prevent the formation of singularitics, even without surface tension; i.e., small smooth data which satisfy certain natural compatibility conditions, smooth solutions exist for all time. Uniqueness of the solution for any finite time interval is also proved.     

Simplified renormalization group method for ordinary differential equations

The renormalization group (RG) method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it. This article investigates higher order RG equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. It is shown that the higher order RG equation maintains the similar theorems to those provided by the first order RG equation, which are theorems on well-definedness of approximate vector fields, and on inheritance of invariant manifolds from those for the RG equation to those for the original equation, for example. Since the higher order RG equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that this simplified RG equation is sufficient to determine whether the trivial solution to time-dependent linear equations is hyperbolically stable or not, and thereby a synchronous solution of a coupled oscillators is shown to be stable.     

A Characteristic Mixed Finite Element Two-Grid Method for Compressible Miscible Displacement Problem

A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. The concentration equation is treated by a mixed finite element method with characteristics (CMFEM) and the pressure equation is treated by a parabolic mixed finite element method (PMFEM). Two-grid algorithm is considered to linearize nonlinear coupled system of two parabolic partial differential equations. Moreover, the $L^q$ error estimates are conducted for the pressure, Darcy velocity and concentration variables in the two-grid solutions. Both theoretical analysis and numerical experiments are presented to show that the two-grid algorithm is very effective.     

A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem . We show that for some classes of function we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity. Received July 9, 1998 / Published online March 16, 2000     

Controllability and influence of the spatial discretization of the beam equation

We study the problem of the control of a finite difference semi-discrete scheme for the 1D beam equation modelling the transverse vibrations of a beam with fixed ends. We show that for most numerical schemes, due to high frequency spurious solutions, the observability may be lost under numerical discretizations when the mesh size tends to zero. We then prove that the uniform observability property is obtained by truncating the high frequencies. We also show how spatial discretization can be performed by using our nonstandard finite difference scheme.     

A stable boundary elements method for magnetohydrodynamic channel flows at high Hartmann numbers

The article is devoted to extension of boundary element method (BEM) for solving coupled equations in velocity and induced magnetic field for time dependent magnetohydrodynamic (MHD) flows through a rectangular pipe. The BEM is equipped with finite difference approach to solve MHD problem at high Hartmann numbers up to 10⁶. In fact, the finite difference approach is used to approximate partial derivatives of unknown functions at boundary points respect to outward normal vector. It yields a numerical method with no singular boundary integrals. Besides, a new approach is suggested in this article where transforms 2D singular BEM's integrals to 1D nonsingular ones. The new approach reduces computational cost, significantly. Note that the stability of the numerical scheme is proved mathematically when computational domain is discretized uniformly and Hartmann number is 40 times bigger than length of boundary elements. Numerical examples show behavior of velocity and induced magnetic field across the sections.     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.     

Nilpotent elements in group rings

The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.This work has been supported by N.R.C. Grant No. A-5300.     

Natural convection in shallow water

Starting from the 3D Boussinesq model and taking the limit as the domain thickness tends to zero, we derive rigorously the 2D model for natural convection in shallow water. The model reduces to a degenerated elliptic equation for the pressure, an explicit formula for horizontal components of the velocity and a vertical diffusion for the vertical component. The macroscopic flow is driven by temperature variations as well as the bottom topography.     

速度追踪问题中鞍点系统的新分裂迭代

 有限元离散一类速度追踪问题后得到具有鞍点结构的线性系统,针对该鞍点系统,本文提出了一种新的分裂迭代技术.证明了新的分裂迭代方法的无条件收敛性,详细分析了新的分裂预条件子对应的预处理矩阵的谱性质.数值结果验证了对于大范围的网格参数和正则参数,新的分裂预条件子在求解有限元离散速度追踪问题得到的鞍点系统时的可行性和有效性.     

A finite volume element formulation and error analysis for the non-stationary conduction–convection problem

The non-stationary conduction–convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element (FVE) method. A fully discrete FVE formulation and the error estimates between the fully discrete FVE solutions and the accuracy solution are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the FVE formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non-stationary conduction–convection problem.     

关于半局部群环

Let R be an associative ring with identity.R is said to be semilocal if R/J(R)is(semisimple)Artinian,where J(R)denotes the Jacobson radical of R.In this paper,we give necessary and sufficient conditions for the group ring RG to be semilocal,where G is a locally finite nilpotent group.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

Guaranteed velocity error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L². Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart‐Thomas discretization which is related to the Crouzeix‐Raviart nonconforming finite element scheme in the lowest‐order case. The effective and guaranteed a posteriori error control for this nonconforming velocity‐oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf‐sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1411–1432, 2016