Description du projet :
Numerical methods for fully nonlinear equations are now gathering a lot of attention
from computational scientists. Numerical techniques range from viscosity solutions based methods, higher order finite elements, finite differences, or variational approaches.
The first application for the validation of all these methods is usually the prototypical Monge-Ampère equation. The starting point of this project is the numerical framework based on several variational approaches developed by the PI and co-authors based on least-squares or augmented Lagrangian methods.
These variational methods are designed to be applied to some elliptic fully nonlinear equations, starting with the Monge-Ampère equation, but also for some Eikonal and prescribed Jacobian equations. This project will build on this framework to develop several new algorithms, together with the corresponding theoretical convergence investigations, mainly in two directions:
1) Extension of the variational approaches. Apply the framework to a large variety of first or second order fully nonlinear elliptic equations, including Pucci's, vectorial
Eikonal equations, prescribed Jacobian equation, or the three-dimensional
Extend the methodology to the numerical approximation of the elliptic prescribed
Jacobian inequalities, and to the evolutive parabolic Monge-Ampère equation.
Study the mathematical convergence properties of the existing least-squares approach.
2) Design of adaptive finite element methods. In order to address non-smooth
problems, tackle the design of adative and mesh refinement techniques of the finite
element discretization in space of the Dirichlet problem for the Monge-Ampère equation in two dimensions of space.
In that framework, those methods are related to adaptive methods for nonlinear Stokes problems and heuristics. Extend the analysis to Eikonal equations.
Research team within HES-SO:
Makhlouf Shabou Basma
, Caboussat Alexandre
Partenaires académiques: 516,International Business Management
Durée du projet:
01.01.2017 - 31.12.2020
Montant global du projet: 394'554 CHF