Description du projet
De nouvelles approximations permettent l’application de puissants solveurs conventionnels
Les équations différentielles partielles (EDP) sont fondamentales pour décrire le monde qui nous entoure. Elles expliquent mathématiquement comment un phénomène donné évolue en fonction des changements affectant d’autres facteurs. Elles peuvent modéliser les dynamiques liées aux échanges gazeux dans le système circulatoire, aux variations du marché boursier et à la propagation des rayons X dans les matériaux. Les EDP entièrement non linéaires constituent un cas particulièrement délicat pour lequel les solutions conventionnelles peuvent ne pas exister ni même être définies. Le projet DAFNE, financé par l’UE, jettera les bases de la mise en œuvre d’approches de solution utilisant de puissantes méthodes d’éléments finis ayant un impact dans des domaines allant de la physique et de la géométrie aux phénomènes de transport et à la finance.
Objectif
Fully nonlinear partial differential equations (PDE) arise in many applications ranging from physics to economy. They are different from PDEs in mechanics, and the PDE theory relies on the generalized solution concept of so-called viscosity solutions. Monotone finite difference methods (FDM) are provably convergent for approximating viscosity solutions, but are restricted to regular meshes and low-order approximations, thus having limitations in resolving realistic geometries or dealing with local mesh refinement. As viscosity solutions are lacking smoothness properties in general, adaptive approximations are desirable. In contrast to FDM, finite element methods (FEM) offer the possibility of high-order approximations with flexibility in adaptive and automatic mesh design. However, provably convergent FEM formulations for viscosity solutions to nonvariational problems are as yet unknown.
With a background in the numerical analysis of PDEs, especially the theory of FEM and adaptive algorithms, DAFNE aims at laying the theoretical and practical foundation for the application of FEM and automatic mesh-refinement algorithms to fully nonlinear equations. The focus is on the large class of Hamilton-Jacobi-Bellman (HJB) equations. They originated from stochastic control problems, but more generally comprise many classical and relevant equations like Pucci's equation or the Monge-Ampère equation
with applications in finance, optimal transport, physics, and geometry.
The novel approach is to estimate local regularity properties through the control variable in the HJB formulation. This (a) gives rise to new regularization strategies and (b) indicates where the mesh needs to be refined. Both achievements are key to the design of a new FEM formulation.
The project is at the frontiers of PDE analysis, numerical analysis, and scientific computing. The long-term goal is to establish the first convergence proofs for adaptive FEM simulations of fully nonlinear phenomena.
Champ scientifique
- natural sciencescomputer and information sciencescomputational science
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesmathematicsapplied mathematicsnumerical analysis
Mots‑clés
Programme(s)
Thème(s)
Régime de financement
ERC-STG - Starting GrantInstitution d’accueil
07743 JENA
Allemagne