Edward J. Dean

A wave equation approach to the numerical solution of the Navier-Stokes equations for incompressible viscous flow

Abstract In this Note we describe a novel method for the solution of the Navier-Stokes equations for incompressible viscous fluids. This method, which can be viewed as an alternative to the methods of characteristics, takes advantage of a time discretization by operator splitting to decouple incompressibility-diffusion from advection. The incompressibility-diffusion steps can be treated by classical Stokes solvers. Concerning the advection steps, thanks to the incompressibility of the advecting field, we can replace the corresponding transport equations by second order in time wave equations, which are much easier to solve numerically despite the fact that they are associated to degenerate elliptic operators. Numerical experiments confirm the good computational properties of the new method.

Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach

The main goal of this Note is to discuss a method for the numerical solution of the two-dimensional elliptic Monge–Ampere equation with Dirichlet boundary conditions (the E-MAD problem). This method relies on the reformulation of E-MAD as a problem of Calculus of Variation involving the biharmonic operator (or closely related operators), and then to a saddle-point formulation for a well-chosen augmented Lagrangian functional, leading to iterative methods such as Uzawa–Douglas–Rachford. The above methodology applies to problems other than E-MAD (such as the Pucci equation). The results of numerical experiments are presented. They concern the solution of E-MAD on the unit square (0,1)×(0,1); the first test problem has a known smooth closed form solution which is easily computed with optimal order of convergence. The second test problem has also a known closed form solution; the fact that this solution has the H2(Ω)-regularity, but not the C2(Ω) one, does not prevent optimal order of convergence. Finally, the third test problem having no smooth solution is more costly to solve and leads to discrete solutions showing negative curvature near the corners. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation

The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge-Ampère equation. The results of numerical experiments will illustrate the capabilities of these methods.

Supercomputer solutions of partial differential equation problems in computational fluid dynamics and in control

Abstract This paper discusses solution methods for partial differential equation problems occuring in computational fluid dynamics (CFD) and in control. The problems considered here are (i) the numerical simulation of incompressible viscous flows modeled by the Navier-Stokes equations and (ii) a boundary control problem for the wave equation. Both problems are approximated by a combination of finite element methods for the space discretization and finite difference methods for the time discretization. To achieve the solution, those methods are combined with preconditioned conjugate gradient algorithms; the accurate solution of the control problem also requires an additional Tychonoff regularization procedure.

An Approximate Factorization/Least Squares Solution Method for a Mixed Finite Element Approximation of the Cahn-Hilliard Equation

We discuss in this article the numerical solution of the Cahn-Hilliard equation modelling the spinodal decomposition of binary alloys. The numerical methodology combines a second-order finite difference time discretization with a mixed finite element space approximation and a least squares formulation based on an approximate factorization of a fourth-order elliptic operator which appears in the numerical model. The least squares problem—which is linear—is solved by a preconditioned conjugate gradient algorithm. The results of numerical experiments illustrate the possibilities of the methods discussed in this article.

APPLICATIONS OF OPERATOR SPLITTING METHODS TO THE NUMERICAL SOLUTION OF NONLINEAR PROBLEMS IN CONTINUUM MECHANICS AND PHYSICS

The main goal of this paper is to describe operator splitting methods for the solution of time dependent differential equations, andto discuss their application to the numerical solution of nonlinear problems such as the Navier-Stokes equations for incompressible viscous fluids, the linear eigenvalue problem, the Hartree equation for the Helium atom, and finally to the solution of a non convex problemfrom the calculus of variations associated to the physics of liquid crystals. Numerical results will be presented showing the potential of such methods.

An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newtons method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.

An operator splitting approach to multilevel methods

In this article, we show that multigrid-like algorithms can be obtained by combining space decomposition with time discretization by operator splitting.

A model trust-region modification of Newton's method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is Newtons method for a nonlinear differential operator equation. A model trust-region approach to globalizing the quasilinearization algorithm is presented. A double-dogleg implementation yields a globally convergent algorithm that is robust in solving difficult problems.

Application of Exact Controllability to the Computation of Scattering Waves

The main goal of this article is to discuss an application of the Hilbert Uniqueness Method of J.L. Lions to the simulation of the scattering of planar waves by either perfectly conducting or coated materials. The methodology combines a time-dependent formulation, optimal control, finite-difference and finite-element discretizations, domain decomposition techniques to handle the coating layer and a preconditioned conjugate gradient algorithm to solve the exact controllability problem. The numerical experiments presented here are related to the test cases of the 2nd Conference/Workshop on Approximation and Numerical Solution of the Maxwell Equations, Washington, D.C., November 1993. The results are in good agreement with those obtained by quite different methods.