Periodic Reporting for period 2 - DAFNE (Discretization and adaptive approximation of fully nonlinear equations)
Periodo di rendicontazione: 2023-01-01 al 2024-06-30
The project is concerned with the numerical approximation of fully nonlinear partial differential equations (PDEs). In general, PDEs are fundamental to our description of the world around us and the approximation of solutions is fundamental for most applications in science and technology. Fully non-linear PDEs are a subclass where conventional methods, for example from computational mechanics, cannot be directly applied because such methods are not tailored to the corresponding generalized solution concepts. But the demand for computing solutions is high, as fully nonliear PDEs are encountered in the fields of geometry, optics, and finance, to name a few. The goal of this project is to develop numerical methods that provably approximate the solution for prototypical model problems. The main challenges include the design and implementation of such methods, the mathematical analysis of their convergence, and the computational efficiency. The latter means that an approximation of a given quality should be found with minimal computational resources. Algorithmically, this means that the computer model should be adaptive in a feedback loop on the basis of the result previously computed.
So far, we have focussed on two model problems: (A) the linear elliptic problem in nondivergence form and (B) the Monge-Ampère equation in two dimensions as a nonlinear example.
For the linear problem (A), in “Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form” we have developed a new approach based on a tool from analysis, called Alexandrov-Bakelman-Pucci principle. Especially in three dimensions, this is a new methodology that makes it possible to numerically solve a large class of linear problems that previously were not accessible with the finite element method.
For the Monge-Ampère equation (B) we have designed a finite element method based on regularization and have proved its convergence without assuming unrealistic regularity properties of the exact solution. The results are documented in “Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation”. With the new scheme we are able to exploit the great potential of high-order approximations, but for irregular solutions this requires a strategy for error control and mesh adaptation. The first a posteriori error bound was published in our work “Stability and guaranteed error control of approximations to the Monge--Ampère equation” where we also experimentally test the performance of an adaptive scheme based on the new bound.
For the linear problem (A), in “Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form” we have developed a new approach based on a tool from analysis, called Alexandrov-Bakelman-Pucci principle. Especially in three dimensions, this is a new methodology that makes it possible to numerically solve a large class of linear problems that previously were not accessible with the finite element method.
For the Monge-Ampère equation (B) we have designed a finite element method based on regularization and have proved its convergence without assuming unrealistic regularity properties of the exact solution. The results are documented in “Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation”. With the new scheme we are able to exploit the great potential of high-order approximations, but for irregular solutions this requires a strategy for error control and mesh adaptation. The first a posteriori error bound was published in our work “Stability and guaranteed error control of approximations to the Monge--Ampère equation” where we also experimentally test the performance of an adaptive scheme based on the new bound.
All the results obtained so far advance the state of the art. They are the first provably converging finite element methods for the model problems in their generality and are accompanied with reliable a posteriori error bounds. Notably, we do not assume unrealistic regularity properties of the solution but rather work with the appropriate generalized solution concepts. In the forthcoming period, we will explore to what extend our approach is applicable to a larger class of model problems, such as the computation convex envelopes or dynamic control problems.