heywood ranacher navier-stokes
121704424
Finite element approximation of the nonstationary Navier Stokes problem. I Regularity of solutions and second order error estimates for spatial discretization
This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier Stokes equations. Second order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as$t \to 0$ and as $t \to \infty It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.
John Groves Heywood Rolf Rannacher
9713114
Finite element approximation of the nonstationary Navier Stokes problem, part II: Stability of solutions and error estimates uniform in time
In this paper, assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary Navier Stokes problem, for the purpose of extending local a priori error estimates, and local a posteriori error estimates, globally in time.
John Groves Heywood Rolf Rannacher
122286533
Finite element approximations of the nonstationary Navier Stokes problem. Part IV: error estimates for second order time discretization
This paper provides an error analysis for the Crank Nicolson method of time discretization applied to spatially discrete Galerkin approximations of the nonstationary Navier Stokes equations. Second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution. For approximations of an exponentially stable solution, these local error estimates are extended uniformly in time as \text{t} \to \infty
John Groves Heywood Rolf Rannacher
120711219
ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS
Fluid dynamical problems are often conceptualized in unbounded domains. However, most methods of numerical simulation then require a truncation of the conceptual domain to a bounded one, thereby introducing artificial boundaries. Here we analyse out experience in choosing artificial boundary conditions implicitly through the choice of variational formulations. We deal particularly with a class of problems that involve the prescription of pressure drops and/or net flux conditions.
John Groves Heywood Rolf Rannacher Stefan Turek
122652047
Finite element approximation of the nonstationary Navier Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization
This paper continues our error analysis of finite element Galerkin approximation of the nonstationary Navier Stokes equations. Optimal order error estimates, both local and global, are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution. These assumptions take into account the loss of regularity at $t 0$ that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem.
John Groves Heywood Rolf Rannacher
120567285
The exterior nonstationary problem for the Navier Stokes equations
John Groves Heywood
121882038
On stationary solutions of the Navier Stokes equations as limits of nonstationary solutions
John Groves Heywood
221640651
Existence of a Stationary Navier Stokes Flow Past a Rigid Body, with Application to Starting Problem in Higher Dimensions
We consider the large time behavior of the Navier Stokes flow past a rigid body in \mathbb{R}^n$ with $n\geq 3$ We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time $t\rightarrow\infty$ in $L^q$ with $q\in[n,\infty] This is called Finn's starting problem and the three dimensional case was affirmatively solved by Galdi, Heywood and Shibata (1997) The present paper extends their result to the case of higher dimensions. Even for the three dimensional case, our theorem provides new convergence rate, that is determined by the summability of the stationary solution at infinity and seems to be sharp.
Tomoki Takahashi
54854683
An error estimate uniform in time for spectral Galerkin approximations of the Navier Stokes problem.
John Groves Heywood
121357351
A Numerically Based Existence Theorem for the Navier Stokes Equations
We have recently experimented with a new spectral code for the spatially periodic nonstationary Navier Stokes equations, finding a plethora of apparently steady, periodic and chaotic solutions which seem interesting from the point of view of dynamical systems theory. This has stimulated our interest in the general problem of proving the existence of strict solutions of differential equations, that correspond to the computational results of numerical experiments. In this paper we analyze the numerical results of one of our experiments. The computational results appear to approximate what seems to be an unstable steady solution. Our objective is to provide a rigorous a posteriori analysis, proving that there does indeed exist a corresponding close by strict solution, and that it is steady. Our method is based on showing that the computed solution satisfies criteria which imply the convergence of a fixed point iteration, in appropriate function spaces for an existence theorem, using the computed solution as the starting value
John Groves Heywood Wayne Nagata Wenzhe Xie