The aim of this project is to develop, analyze, and implement the finite element method for fully nonlinear second order partial differential equations (PDEs). The research is based on a recent discovery of the PI that Lagrange finite element methods and discontinuous Galerkin methods can be used to approximate the Monge-Ampere equation, the prototypical fully nonlinear second order PDE. As these methods are simple to implement, the computation of the highly nonlinear problem can be performed efficiently and accurately. The project will expand on these results to obtain simple, efficient, yet accurate numerical schemes for a general class of fully nonlinear equations. In addition, the PI will develop and analyze various discretization methods including mixed finite element methods, local discontinuous Galerkin methods, hybridizable Galerkin methods, and Petrov Galerkin methods.
Mathematical modeling plays a key role in the investigation and understanding of many phenomena occurring in the natural sciences, the social sciences and engineering. Yet even for simple problems, closed form solutions are unavailable, and therefore their numerical approximations are the only viable option. As the problems become ever more complex, the need for novel computational methods and innovative analysis becomes imperative to put the United States in the forefront in science and engineering. The class of problems studied in this project arise in numerous mathematical modeling applications including weather phenomena, determining the initial shape of the universe, optimal reflector design, differential geometry, optimal transport, mathematical finance, image processing, and mesh generation. Despite their significance in the physical sciences and pure and applied mathematics, the numerical approximation of these problems remains a relatively untouched area. Therefore, there is a growing need to develop accurate schemes for these types of equations. As progress of solving any of these application problems largely depends on progress of solving their governing equations, and since numerical methods for these equations are still in their infancy, any progress in the design, implementation, and convergence analysis will have an immediate impact in advancing these application areas.
The objective of this project was to develop computational methods and analytical tools for a specific classof differential equations. The types of problems studied appear in several application models of engineering, meteorology, finance, cosmology differential geometry, and theoretical mathematics. Except in simple and unrealistic settings, these differential equations cannot be solved analytically; rather, a computational algorithm is required to construct approximate discrete solutions. While the motivation to develop computational methods for the class of problems is driven by the above application areas, the main focus of project lies in the analysis of the methods and the development of rigorous mathematical tools to sharply measure the error of the approximate solutions. Such a program achieves two purposes: first, the analysis provides a guaranteed criterion to compute a solution within a given error tolerance; second, the mathematics gives insight of the properties of the computational methods, which generally leads to less costly and faster algorithms. The specific outcomes of this project consistedof two integrated components. First, the PI developed new types of methods for fully nonlinear differential equations and has disseminated the results through peer--reviewed research articles and research talks.The methods are relatively simple to implement and are supported on existing computational software. As a result, practitioners from other disciplines may readily implement the methods on current software libraries. In addition, the PI has derived sharp bounds of the error of the approximate problem in relation to the complexity of the (discrete) problem. The second component of the project integrated the topics and contents of the material into graduate--level courses and graduate theses at the University of Pittsburgh. The PI advised two master's students, both from underrepresented groups, with subjects related to the project.
