Project description
Novel approximations enable application of powerful conventional solvers
Partial differential equations (PDEs) are fundamental to our description of the world around us, explaining mathematically how a given phenomenon changes with respect to changes in other factors. They can model dynamics related to gas exchange in the circulatory system, changes in the stock market and the propagation of X-rays in materials. Fully non-linear PDEs are a particularly tricky case for which conventional solutions may not exist or even be defined. The EU-funded DAFNE project will lay the foundations to implement solution approaches using powerful finite element methods with impact in areas from physics and geometry to transport phenomena and finance.
Objective
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.
Fields of science
- natural sciencescomputer and information sciencescomputational science
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesmathematicsapplied mathematicsnumerical analysis
Keywords
Programme(s)
Topic(s)
Funding Scheme
ERC-STG - Starting GrantHost institution
07743 JENA
Germany