Problems in the mathematical theory of finite element methods

Data assimilation algorithms improve the predictive capability of computational models for physical systems by incorporating a finite number of measurements of the system into the model. Such measurements might for example come from sensors in an aircraft wing or imaging of blood flow. This project will develop and analyze optimal data assimilation algorithms in cases where a physical law describing behavior inside of a body of interest is known but knowledge of the system’s state on the boundary is unknown. Because full knowledge of boundary behavior is needed to make a precise prediction, the resulting models yield imprecise knowledge of the system’s behavior. In order to partially compensate for lack of boundary data, physical measurements are thus assimilated into the model in order to narrow the range of possible outcomes. The techniques that will be developed in this project have potential applications in areas such as aircraft design and biotechnology. The project will contribute to graduate training by supporting Ph.D. dissertation work. PDE-based optimal recovery addresses situations in which insufficient PDE problem data is present to guarantee a unique solution. This project will focus on solution to elliptic PDE with boundary data that is missing for reasons such as unknown physics, unknown boundary values, or modification of boundary data in numerical schemes. Compensating information is provided in the form of a finite set of measurements of the unknown solution. The recovery framework that is employed defines a notion of optimal recovery and provides a finite element algorithm for its approximation. This algorithm involves discrete solution of a set of fractional (nonlocal) elliptic problems posed on the boundary of the domain. These fractional problems are solved using an established sinc quadrature-based approximation scheme which requires solving a set of second-order reaction-diffusion problems, some of which are singularly perturbed. The project will provide a priori and a posteriori error analysis for relevant reaction-diffusion and fractional-order problems on surfaces and then incorporate these estimates into improved a priori and new a posteriori error estimates for the optimal recovery problem. The main focus of the error analyses will be global and local maximum norms, which are natural in the context of the target optimal recovery problem. 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.