Martine Marion

Nonlinear Galerkin methods

This article presents a new method of integrating evolution differential equations—the non-linear Galerkin method—that is well adapted to the long-term integration of such equations.While the usual Galerkin method can be interpreted as a projection of the considered equation on a linear space, the methods considered here are related to the projection of the equation on a nonlinear manifold. From the practical point of view some terms have been identified as small, and sometimes.(but not always) disregarded.

Error estimates on a new nonlinear Galerkin method based on two-grid finite elements

A new nonlinear Galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations. The new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size H and one fine grid with grid size

Attractors for reaction-diffusion equations: existence and estimate of their dimension

h \ll H

Nonlinear Galerkin methods: the finite elements case

. Nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time. With linear finite element discretizations, it is proved that the difference between the new nonlinear Galerkin solution and the standard Galerkin solution in

On the rate of convergence of the nonlinear Galerkin methods

H^1 (\Omega )

Approximate inertial manifolds for reaction-diffusion equations in high space dimension

norm is of the order of

A class of numerical algorithms for large time integration: the nonlinear Galerkin methods

H^3

Inertial manifolds associated to partly dissipative reaction-diffusion systems

.

Mathematical Analysis and Optimization of Infiltration Processes

In this paper, we study some questions related to attractors for two types of reaction-diffusion equations : an equation with a polynomial growth nonlinearity and systems admitting a positively invariant region. For these problems, we prove the existence of a maximal attractor which describes the long-time behaviour of the solutions and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our results are applied to several classical equations.

Some remarks on turbulent combustion from the attractor point of view

SummaryWith the increase in the computing power and the advent of supercomputers, the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem. In this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements. The algorithm that we propose, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases. Beside a detailed description of the algorithm, the article includes some technical results on finite elements spaces, and a full study of the stability and convergence of the method.