William J. Layton

A Two-Level Method with Backtracking for the Navier--Stokes Equations

We consider a two-level method for resolving the nonlinearity in finite element approximations of the equilibrium Navier--Stokes equations. The method yields L2 and H1 optimal velocity approximations and an L2 optimal pressure approximation. The two-level method involves solving one small, nonlinear coarse mesh system, one Oseen problem (hence, linear with positive definite symmetric part) on the fine mesh, and one linear correction problem on the coarse mesh. The algorithm we study produces an approximate solution with the optimal, asymptotic in h, accuracy for any fixed Reynolds number. We do not consider the behavior of the error for fixed h as

Introduction to the Numerical Analysis of Incompressible Viscous Flows

Re\rightarrow \infty

A two-level discretization method for the Navier-Stokes equations☆☆☆

, i.e., for flows in transition to turbulence.

Large Eddy Simulation

This book treats the numerical analysis of finite element computational fluid dynamics. Assuming minimal background, the text covers finite element methods; the derivation, behavior, analysis, and numerical analysis of Navier Stokes equations; and turbulence and turbulence models used in simulations. Each chapter on theory is followed by a numerical analysis chapter that expands on the theory. The chapters contain numerous exercises. Introduction to the Numerical Analysis of Incompressible Viscous Flows provides the foundation for understanding the interconnection of the physics, mathematics, and numerics of the incompressible case, which is essential for progressing to the more complex flows not addressed in this book (e.g., viscoelasticity, plasmas, compressible flows, coating flows, flows of mixtures of fluids, and bubbly flows). With mathematical rigor and physical clarity, the book progresses from the mathematical preliminaries of energy and stress to finite element computational fluid dynamics in a format manageable in one semester. Audience: This unified treatment of fluid mechanics, analysis, and numerical analysis is intended for graduate students in mathematics, engineering, physics, and the sciences who are interested in understanding the foundations of methods commonly used for flow simulations.

Two-level Picard and modified Picard methods for the Navier-Stokes equations

Abstract We propose and analyze a two level method of discretizing the nonlinear Navier-Stokes equations. This method has optimal accuracy and requires neither iteration nor the solution of more than a very small number of nonlinear equations.

APPROXIMATION OF THE LARGER EDDIES IN FLUID MOTIONS II: A MODEL FOR SPACE-FILTERED FLOW

At high Reynolds number the fluid velocity is exponentially sensitive to perturbations of the problem data. This sensitivity, however, is not uniform. The large structures (large eddies) evolve deterministically and are thus not sensitive [BFG02]. The small eddies are sensitive because they have a random character.

A multilevel mesh independence principle for the Navier-Stokes equations

Abstract Iterative methods of Picard type for the Navier-Stokes equations are known to converge only for quite small Reynolds numbers. However, we study methods involving just one such iteration at general Reynolds numbers. For the initial approximation a coarse mesh of width h0 is used. The corrected approximation is computed by just one Picard or modified Picard step on a fine mesh of width h1. For example, h1 may be of order O (h 0 2 ) when linear velocity elements are used. The resulting method requires the solution of a (small) system of nonlinear equations on the coarse mesh and only one (larger) linear system on the fine mesh. This two-level Picard method is proven to converge for fixed Reynolds number as h → 0. Further, the fine mesh solution satisfies a quasi-optimal error bound. (The error constants grow as Re → ∞, as for the usual finite element method.) One very heuristic explanation why one step of the (divergent) Picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower-order derivatives; thus they are approximated with higher accuracy on the coarse mesh. This is linked with a “smoothing property” of the fine mesh step.

A defect-correction method for the incompressible Navier-Stokes equations

We present two modifications of continuum models used in large eddy simulation. The first modification is a closure approximation which better attenuates small eddies. The second modification is a change in the boundary conditions for the large eddy model from strict adherence to slip with resistance. For model consistency, the resistance coefficient is a function of the averaging radius so that the models boundary conditions reduce to a no-slip as the averaging radius decreases to zero.

Numerical Solution of the Stationary Navier--Stokes Equations Using a Multilevel Finite Element Method

Multilevel, finite element discretization methods for the Navier–Stokes equations are considered. In contrast to usual multilevel methods, a superlinear scaling of the consecutive meshwidths

Error analysis for finite element methods for steady natural convection problems

h_J + 1 = \mathcal {O}(h_j^{\alpha (j)} )