Approximation and Analysis of Selected Nonsmooth, Nonlinear, and Nonlocal Equations

The numerical approximation of partial differential equations, and the analysis of schemes to approximate the solution of classical models in the pure and applied sciences, is a well-established topic. There are general theories to deduce convergence and accuracy of approximations. However, new classes of models have recently appeared that do not lend themselves to the general treatment and require new techniques and ideas.

In this project the PI aims to develop, together with students and a postdoctoral associate, rigorous analyses of approximation techniques for nonsmooth, nonlinear, and nonlocal systems that describe a wide range of phenomena. The analysis will require a careful interplay between subtle smoothness properties of the solutions and the fine structure of the problems and schemes at hand.

The analysis will borrow techniques, not among the standard tools invoked in numerical analysis, from other fields of mathematics. The research will enhance modeling and prediction capabilities for this important class of models, and early-career researchers will be trained through involvement in the project.

Systems under study in this project include a) initial value problems where the presence of singular data or the inherent nature of the equation make the solution very rough; b) nonlinear systems where the strength of the nonlinearity is such that even for smooth data one cannot immediately assert the smoothness of the solution; c) nonlocal problems, those which describe long range interactions or memory effects and, thus, the determination of the state of the system at one point requires global knowledge; and d) nonvariational equations, that is, those that do not arise from conservation principles. In all these examples, standard arguments invoked to establish stability and convergence of numerical methods are not effective.

As an outcome of this work, new numerical techniques will be developed, and the existing ones will be strengthened by solid mathematical analysis of their approximation properties.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.