John G. Heywood

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

Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization

t \to 0

ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

and as

Finite element approximation of the nonstationary Navier-Stokes problem, part III. Smoothing property and higher order error estimates for spatial discretization

t \to \infty

Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time

. 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.

On the question of turbulence modeling by approximate inertial manifolds and the nonlinear Galerkin method

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

On The Steady Transport Equation

{\text{t}} \to \infty

Theory of the Navier-Stokes equations

.

Contributions to Current Challenges in Mathematical Fluid Mechanics

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.

On the Existence and Uniqueness Theory for Steady Compressible Viscous Flow

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