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DAFNE

Financiado

Discretization and adaptive approximation of fully nonlinear equations

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... ver más

30/06/2026

UNI JENA

1M€

Presupuesto del proyecto: 1M€

Líder del proyecto

FRIEDRICHSCHILLERUNIVERSITT JENA No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Ver 2 Participantes

Financiación concedida El organismo H2020 notifico la concesión del proyecto el día 2020-10-22

Línea de financiación objetivo El proyecto se financió a través de la siguiente ayuda:

ERC-2020-STG: ERC STARTING GRANTS

I+D

Cerrada hace 5 años

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2 Participantes

UNI JENA

1.45M€ | Lider

SOFTENG,S.L.

510.32K€ | Participante

Últimas noticias

30-11-2024: Cataluña Gestión For... Se ha cerrado la línea de ayuda pública: Gestión Forestal Sostenible para Inversiones Forestales Productivas para el organismo:

29-11-2024: IDAE En las últimas 48 horas el Organismo IDAE ha otorgado 4 concesiones

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Duración del proyecto: 68 meses Fecha Inicio: 2020-10-22
Fecha Fin: 2026-06-30

Líder del proyecto

FRIEDRICHSCHILLERUNIVERSITT JENA

TRL 4-5

Presupuesto del proyecto 1M€

Descripción del proyecto 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.

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