Numerical Methods and Algorithms for Second Order Fully Nonlinear Partial Differential Equations

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Second order fully nonlinear partial differential equations (PDEs) arise from many areas in science and engineering such as differential geometry, optimal control, mass transportation, materials science, meteorology, geostrophic fluid dynamics. They constitute the most difficult class of differential equations to analyze analytically and to approximate numerically. In the past two decades, enormous advances in the theoretical analysis has been achieved, based on the viscosity solution theory, for second order fully nonlinear PDEs. On the other hand, in contrast to the success of the PDE analysis, numerical solutions for general second order fully nonlinear PDEs is mostly an untouched area, and computing viscosity solutions of second order fully nonlinear PDEs has been impracticable. In this research project, the PI plans to conduct an extensive and systematic study of numerical methods and algorithms for second order fully nonlinear PDEs based on a newly developed moment solution concept and a constructive vanishing moment methodology. The specific tasks of the project include (i) to continue developing the moment solution theory for Monge-Ampere type PDEs and for general second order fully nonlinear elliptic and parabolic PDEs in two and three dimensions; (ii) to develop finite element, mixed finite element, discontinuous Galerkin, and spectral Galerkin discretization methods; (iii) to analyze convergence and rates of convergence for all proposed discretization methods; (iv) to design preconditioned Newton type nonlinear solvers; (v) to develop computer code based on Comsol Multiphysics platform for implementing the proposed discretization methods and solution algorithms on high performance workstations.The completion of the proposed project will have a profound impact on both theoretical study and numerical approximations of second order fully nonlinear PDEs. It will provide the first practical and successful methodology/approach, which is backed by rigorous PDE and numerical theories, for approximating second order fully nonlinear PDEs. As a by-product, the moment solution theory will enrich the current understanding of the viscosity solution theory, and might be very likely to provide a logical and natural generalization/extension for the viscosity solution concept. The findings of the proposed research will provide the much needed capability and enabling tools for computing correctly and efficiently those challenging fully nonlinear PDEs from differential geometry, general relativity, fluid mechanics, materials science, optimal control, mass transportation, meteorology, image processing, especially, in the cases where there are no theories. The educational component of this project is to engage and train graduate students in developing necessary applied and computational mathematics knowledge and skills so that they can pursue a successful career in science and engineering in the future.