Adaptive numerical methods guided by error estimators have become one of the essential technologies for modern numerical approximations of partial differential equations (PDE) and are widely used in scientific and industrial computations for highly challenging problems. The adoption of adaptive methods has been driven by their increased computational efficiency and accuracy, and also by major recent advances on their mathematical analysis through proofs of convergence and of optimality properties.

Recently, the concept of limit spaces has emerged as a foundational tool for proofs of convergence of adaptive algorithms for both challenging nonlinear PDE and nonconforming numerical methods that fall outside the scope of existing theory. This has been demonstrated most recently by the recent proof of convergence of adaptive nonconforming methods for fully nonlinear Hamilton-Jacobi-Bellman (HJB) and Isaacs equations.

This project aims to develop an extensive general theory of limit spaces for nonconforming numerical approximations and apply the theory to the challenging systems of nonlinear PDE from Mean Field Games (MFG). MFG arise in models of stochastic differential games of optimal control, and are of very strong current interest due to their numerous applications in high-priority industrial and economic areas such as renewable energy systems and markets, which are of national and international importance. MFG consist of a nonlinearly coupled system of an HJB equation and a Fokker-Planck equation, and therefore stand as a key problem that stands to benefit from the recent advances in the analysis of limit spaces.

The project will:

[A] lead to a new mathematical theory for nonstandard function spaces, permitting the rigorous analysis of convergence of adaptive numerical methods for a large range of problems.

[B] result in new numerical methods for the accurate and efficient solution of challenging nonlinear PDE of current interest, fully supported by rigorous mathematical theory.

The theoretical advances will be of central importance for the analysis of adaptive algorithms for more general nonlinear PDE and a wider range of numerical methods, and the numerical advances will be of benefit to application areas of Mean Field Games.