About the regularity of solutions for the nonstationary Navier-Stokes systems

The problem of existence of regular (Hölder continuous) solutions for the multidimensional nonstationary Navier- Stokes system is considered. The results are based on explicit pointwise bounds for the velocities. The coercivity estimates in special weighted spaces for the Stokes system are used and the smallness of the Reynolds numbers is assumed.     

Effect of errors in the spatial geometry for temperature dependent Stokes flow

A priori bounds are established for the solution to the problem of Stokes flow in a bounded domain, for a viscous, heat conducting, incompressible fluid, when changes in the spatial geometry are admitted. These bounds demonstrate how the velocity field and the temperature field depend on changes in the spatial geometry and also yield a convergence theorem in terms of boundary perturbations. The results have a direct bearing on an error analysis for a numerical approximation to non-isothermal Stokes flow when the boundary of a complicated domain is approximated by a simpler one, e.g., in the procedure of triangulation combined with finite elements.     

Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier–Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order \(\mathcal O(h^{2})\) in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.     

A posteriori estimates for the Stokes eigenvalue problem

We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a‐posteriori error bounds. The estimates are verified by numerical computations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations

We describe a method to estimate the guaranteed error bounds of the finite element solutions for the Stokes problem in mathematically rigorous sense. We show that an a posteriori error can be computed by using the numerical estimates of a constant related to the so-called inf-sup condition for the continuous problem. Also a method to derive the constructive a priori error bounds are considered. Some numerical examples which confirm us the expected rate of convergence are presented.     

Error bounds for the finite-element approximation of the exterior Stokes equations in two dimensions

In this paper we design high-order (non)local artificial boundaryconditions (ABCs) which are different from those proposed byHan, H. & Bao, W. (1997 Numer. Math., 77, 347–363)for incompressible materials, and present error bounds for thefinite-element approximation of the exterior Stokes equationsin two dimensions. The finite-element approximation (especiallyits corresponding stiff matrix) becomes much simpler (sparser)when it is formulated in a bounded computational domain usingthe new (non)local approximate ABCs. Our error bounds indicatehow the errors of the finite-element approximations depend onthe mesh size, terms used in the approximate ABCs and the locationof the artificial boundary. Numerical examples of the exteriorStokes equations outside a circle in the plane are presented.Numerical results demonstrate the performance of our error bounds.     

A posteriori error estimates for viscous flow problems with rotation

A new functional type a posteriori error estimates for the Stokes problem with rotating term are presented. The estimates give guaranteed upper bounds for the energy norm of the error and provide reliable error indication. Computational properties of the estimates are demonstrated by a number of numerical examples. Bibliography: 37 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 23–34.     

A posteriori error estimates for a compressible Stokes system

A linearized compressible viscous Stokes system is considered. The a posteriori error estimates are defined and compared with the true error. They are shown to be globally upper and locally lower bounds for the true error of the finite element solution. Some numerical examples are given, showing an efficiency of the estimator. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 412–431, 2004.     

A posteriori error estimators for the Stokes equations

Summary We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.This work was accomplished at the Universität Heidelberg with the support of the Deutsche Forschungsgemeinschaft     

A Posteriori Error Estimation of Spectral and Spectral Element Methods for the Stokes/Darcy Coupled Problem

In this paper, we carry out an a posteriori error analysis of Legendre spectral approximations to the Stokes/Darcy coupled equations. The spectral approximations are based on a weak formulation of the coupled equations by using the Beavers-Joseph-Saffman interface condition. The main contribution of the paper consists of deriving a number of posteriori error indicators and their upper and lower bounds for the single domain case. An extension of the upper bounds to the multi-domain case in the spectral element framework is also given.     

Eigenvalue estimates for preconditioned saddle point matrices

New accurate eigenvalue bounds for symmetric matrices of saddle point form are derived and applied for both unpreconditioned and preconditioned versions of the matrices. The estimates enable a better understanding of how preconditioners should be chosen. The preconditioners provide efficient iterative solution of the corresponding linear systems with, for some important applications, an optimal order of computational complexity. The methods are applied for Stokes problem and for linear elasticity problems. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Stabilized finite element method for Navier-Stokes equations with physical boundary conditions

This paper deals with the numerical approximation of the 2D and 3D Navier-Stokes equations, satisfying nonstandard boundary conditions. This lays on the finite element discretisation of the corresponding Stokes problem, which is achieved through a three-fields stabilized mixed formulation. A priori and a posteriori error bounds are established for the nonlinear problem, ascertaining the convergence of the method. Finally, numerical tests are presented, including mesh refinement via error indicators.

     

Residual a posteriori error estimates for a 3D mortar finite-element method: the Stokes system

We consider a non-conforming domain decomposition techniquefor the discretization of the three-dimensional Stokes equationsby the mortar finite-element method. Relying on the velocity–pressureformulation of the system, we perform the numerical analysisof residual error indicators for this problem and we prove thatthe error estimators provide upper and lower bounds for theenergy norm of the mortar finite-element solution.     

On the spectrum of the stokes operator

We prove Li-Yau type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier-Stokes equations.     

Stoxes方程初边值问题的Phragmen－Lindelof二择性原理

本文对非定常的Ｓｔｏｋｅｓ方程的初边值问题证明了Ｐｈｒａｇｍｅｎ－Ｌｉｎｄｅｌｏｆ二择性原理，即证明Ｓｔｏｋｅｓ流函数的能量，随着与带状区域有限端距离的增加必定或者按指数率增长或者按指数率衰减．对能量衰减情况建立了Ｓｔｏｋｅｓ流速度的最大模的点点估计．并提出求全能量上界的方法．     

OPTIMAL MIXED H-P FINITE ELEMENT METHODS FOR STOKES AND NON-NEWTONIAN FLOW

1. IntroductionMOtivated,,by some advalltages of h -- p FEM over the classic FEM uncovered by recelltcomputation works(see-[19]), Schwab and Sari [11] have considered the ~d h--p lhate elementmethod for Non-Newtonian flow based upon a three-field Stokes formulation emallating fromMarization of some dmerellt models of Non-Newtoulan flow, in which, stress, velocity andPressure are coupled. Theoretical analysis and tailored numerical expert~8 show that theabed h -- p finite elemellt method …     

An exponential decay estimate for the stationary axisymmetric perturbation of Poiseuille flow in a circular pipe

We prove a decay estimate for the steady state incompressible Navier-Stokes equations. The estimate describes the exponential decay, in the axial direction of a semi-infinite circular tube, for an energy-type functional in terms of the axisymmetric perturbation of Poiseuille flow, provided that the Reynolds number does not exceed a critical value, for which we exhibit a lower and an upper bound. Since the motion is considered axisymmetric we use a stream function formulation, and the results are similar to those obtained by Horgan [8], for a two-dimensional channel flow problem. For the Stokes problem our estimate for the rate of decay is a lower bound to the actual rate of decay which is obtained from an asymptotic solution to the Stokes equations. Finally we describe a numerical approach to computing bounds to the energy functionalE(0).     

A Posteriori Error Analysis for Mixed Finite Element Solution of the Two-Dimensional Stationary Stokes Problem

In this paper we present a posteriori error estimator in a suitable norm of mixed finite element solution for two-dimensional stationary Stokes problem. The estimator is optimal in the sense that, up to multiplicative constant, the upper and lower bounds of the error are the same. The constants are independent of the mesh and the true solution of the problem.     

一类KKT系统的结构敏度分析

本文讨论一类KKT系统的敏度分析，这类KKT系统产生于用有限元方法离散Stokes方程，有结构特性．首先给出了最佳向后扰动界，接下来定义了偏条件数并导出了表达式．最后给出了新的扰动界．