Roy A. Nicolaides

On finite element methods of the least squares type

Abstract A theoretical framework for the least squares solution of first order elliptic systems is proposed, and optimal order error estimates for piecewise polynomial approximation spaces are derived. Numerical examples of the least squares method are also provided.

A Few Tools For Turbulence Models In Navier-Stokes Equations

Abstract This article is for those who have already a computer program for incompressible viscous transient flows and want to put a turbulence model into it. We discuss some of the implementation problems that can be encountered when the Finite Element Method is used on classical turbulence models except Reynolds stress tensor models. Particular attention is given to boundary conditions and to the stability of algorithms. Introduction Many scientists or engineers turn to turbulence modeling after having written a Navier-Stokes solver for laminar flows. For them turbulence modeling is an external module into the computer program. Generally, the main ingredients to built a good Navier-Stokes solver are known; this includes tools like mixed approximations for the velocity u and pressure p to avoid checker board oscillations and also upwinding to damp high Reynolds number oscillations; however the problems that one may meet while implementing a turbulence model are not so well known because these models have not been studied much theoretically. Judging from the literature [3] [11] [12] [15] [19] [22] the most commonly used turbulence models seem to be algebraic eddy viscosity models (zero equation models) k= e models (two equations models) Reynolds stress models All three start from a decomposition of u and p into a mean part and a fluctuating part u’. However oscillations are understood either as time oscillations or space oscillations or even variations due to changes in initial conditions. In any case, the decomposition u+u’ is applied to the Navier-Stokes equations.

On conforming mixed finite element methods for incompressible viscous flow problems

Abstract The asymptotic rates of convergence for approximate solutions of linearizations of the stationary Navier-Stokes equations are computationally determined for some specific choices of conforming finite element spaces. These rates are computed for norms of physical interest and are compared to available theoretical estimates. It is shown that equivalent rates of convergence are achieved by algorithms which differ greatly in their computer storage and time requirements. The solution of the discrete system of equations resulting from the finite element discretization is also discussed.

Stability of Gaussian elimination without pivoting on tridiagonal Toeplitz matrices

Abstract Using the simple vehicle of tridiagonal Toeplitz matrices, the question of whether one must pivot during the Gauss elimination procedure is examined. An exact expression for the multipliers encountered during the elimination process is given. It is then shown that for a prototype Helmholtz problem, one cannot guarantee that elimination without pivoting is stable.

A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.