Past

In this talk we present different strategies for regularization of the pure Newton method

(minimization problems)and of the Gauss-Newton method (systems of nonlinear equations).

For these schemes, we prove general convergence results. We establish also the global and

local worst-case complexity bounds. It is shown that the corresponding search directions can

be computed by a standard linear algebra technique.

We present a novel enhanced finite element method for the Darcy problem starting from the non stable

continuous $P_1 / P_0$ finite element spaces enriched with multiscale functions. The method is a departure

from the standard mixed method framework used in these applications. The methods are derived in a Petrov-Galerkin

framework where both velocity and pressure trial spaces are enriched with functions based on residuals of strong

equations in each element and edge partition. The strategy leads to enhanced velocity space with an element of

the lowest order Raviart-Thomas space and to a stable weak formulation preserving local mass conservation.

Numerical tests validate the method.

Jointly with Gabriel R Barrenechea, Universidad de Concepcion &

Frederic G C Valentin, LNCC

Strong horizontal gradients of density are responsible for the occurence of a large number of (often catastrophic) flows, such as katabatic winds, dust storms, pyroclastic flows and powder-snow avalanches. For a large number of applications, the overall density contrast in the flow remains small and simulations are carried in the Boussinesq limit, where density variations only appear in the body-force term. However, pyroclastic flows and powder-snow avalanches involve much larger density contrasts, which implies that the inhomogeneous Navier-Stokes equations need to be solved, along with a closure equation describing the mass diffusion. We propose a Lagrange-Galerkin numerical scheme to solve this system, and prove optimal error bounds subject to constraints on the order of the discretization and the time-stepping. Simulations of physical relevance are then shown.

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