Carol S. WOODWARD, Lawrence Livermore National Laboratory, USA:
A reconsideration of fixed point methods for nonlinear systems

Newton-Krylov methods have proven to be very effective for solution of large-scale, nonlinear systems of equations resulting from discretizations of PDEs. However, increasing complexities and newer models are giving rise to nonlinear systems with characteristics that challenge this commonly used method. In particular, for many problems, Jacobian information may not be available or it may be too costly to compute. Moreover, linear system solves required to update the linear model within each Newton iteration may be too costly on newer machine architectures.

Fixed point iteration methods have not been as commonly used for PDE systems due to their slow convergence rate. However, these methods do not require Jacobian information nor do they require a linear system solve. In addition, recent work has employed Anderson acceleration as a way to speed up fixed point iterations.

In this presentation, we will discuss reasons for success of Newton’s method as well as its weaknesses. Fixed point and Anderson acceleration will be presented along with a summary of known convergence results for this accelerated method. Results will show benefits from this method for a number of applications. In addition, the impacts of these methods will be discussed for large-scale problems on next generation architectures.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC. LLNL-ABS-663073.

Video from the talk

Friday, December 4, 10:30 a.m. to 11:30 a.m., Turing lecture hall, building 1, Paris-Rocquencourt center. Coffee from 10:15 a.m.

Roland BECKER, Université de Pau:
Nitsche’s method for incompressible flows

In this talk we study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche’s method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Following the theory of Friedrich’s systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Then we consider further applications of this technique to residual-based stabilization and domain decomposition. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented.

Video from the talk

Tuesday, November 3, 10:30 a.m. to 11:30 a.m., Turing lecture hall, building 1, Paris-Rocquencourt center. Coffee from 10:15 a.m.

Jean-François BABADJIAN, Laboratoire Jacques-Louis Lions, University Paris 6:
Une approche variationnelle de la mécanique de la rupture

Dans cet exposé, nous présenterons un modèle variationnel en mécanique de la rupture fragile introduit par Francfort et Marigo. Celui-ci repose sur une idée originale due a Griffith qui postule l’existence d’une énergie de surface. La propagation des fissures est alors le résultat d’une compétition entre une énergie de volume, l’énergie élastique, et cette énergie de surface. L’approche classique pour étudier ce modèle est basée sur une discrétisation temporelle qui engendre, lorsque le pas de temps tend vers zéro, des solutions faibles en temps continu : le déplacement appartient à un sous espace des fonctions à variation bornée et la fissure est un ensemble rectifiable. Nous montrerons que dans le cas 2D anti-plan, ces solutions faibles sont en fait des solutions fortes au sens ou la fissure est un ensemble fermé en dehors duquel le champ des déplacements est régulier. Enfin nous montrerons comment ce modèle de type frontière/discontinuité libre peut être implémenté numériquement.

Video from the talk

Tuesday, October 6, 10:30 a.m. to 11:30 a.m., Turing lecture hall, building 1, Paris-Rocquencourt center. Coffee from 10:15 a.m.

Enrique D. FERNANDEZ-NIETO, U. of Sevilla, Spain and Tomás MORALES DE LUNA, Cordoba University, Spain:
Some advances on the mathematical modelling and numerical simulation of sediment transport: bedload and suspension

This talk focuses on several aspects related to the matematical modelling and simulation of sediment transport in rivers and coastal areas. Sediment transport is usually classified into bedload and suspension transport. This talk is divided in two parts following these two topics: firstly some recent advances on mathematical models for bedload transport will be presented. Saint-Venant-Exner system is generally used to model the bedload transport in rivers, lakes and coastal areas. We present a deduction of the Saint-Venant-Exner model through an asymptotic analysis of the Navier-Stokes equations.

Secondly we present a model to study suspension transport based on a multi-layer approach. An interesting phenomena related to sediment in suspension are hyperpycnal and hypopycnal plumes: when a river that carries an elevated concentration of suspended sediment comes into the ocean, it can form a plume that advects the sediment from the river mouth which can plunge (hyperpycnal) or float (hypopycnal) in the receiving ambient water. The multilayer approach will allow to accurately describe these phenomena and the vertical distribution of sediment inside the plume.

Video from the talk

Wednesday, September 9, 10:30 a.m. to 11:45 a.m., Turing lecture hall, building 1, Paris-Rocquencourt center. Coffee from 10:15 a.m.