Ellipticity, Optimization and Finite Elements

There are three topics in this proposal: elliptic optimal control problems, elliptic problems with rough coefficients and fully nonlinear elliptic partial differential equations. Ellipticity, optimization and finite elements are central to all of the proposed research projects. The results from the projects in optimal control are relevant for the optimal design processes in engineering. The results from the projects for problems with rough coefficients can be applied to multiscale problems that appear in materials science and geoscience. The results from the projects in fully nonlinear elliptic partial differential equations will provide reliable and useful computational tools for differential geometry and optimal transport. The proposed work will build bridges among the communities of numerical partial differential equations, optimization, elliptic optimal control, multiscale modeling and domain decomposition. The research in elliptic optimal control problems will extend the recent work of the PI and collaborators in distributed control with pointwise state constraints to general cost functions and general partial differential equation (PDE) constraints. It will also develop new error analyses for boundary control problems with control constraints that can be applied to multiscale finite element methods when the coefficients in the PDE constraint are rough. The research in elliptic problems with rough coefficients will develop multiscale finite methods that are based on the local orthogonal decomposition (LOD) methodology with a domain decomposition (DD) twist. It will extend the DD-LOD framework to problems with high contrast channels, to variational inequalities, to Neumann boundary value problems, to fourth order problems and to elliptic boundary control problems with control constraints. The research in fully nonlinear elliptic partial differential equations will focus on problems that involve Monge-Ampere equations: the Minkowski problem, the prescribed Gaussian curvature problem and the second boundary value problem for the Monge-Ampere equation. It is based on novel convexity enforcing finite elements discovered by the PI and collaborators in recent years and a nonlinear least-squares approach. The goal is to develop finite element methods that can capture smooth solutions and that come with a rigorous error analysis and convergence rates. 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.