Fully nonlinear second order partial differential equations (PDEs) are referred to a class of nonlinear second order PDEs which are nonlinear in (at least one) second order partial derivatives of unknown functions. Such a class of PDEs arise from many scientific and engineering fields including astrophysics, differential geometry, geostrophic fluid dynamics, image processing, kinetic theory, materials science, mass transportation, meteorology, and optimal control. They constitute the most difficult class of PDEs to analyze analytically and to approximate numerically. Building on the PI's recent success on developing convergent and efficient numerical methods and algorithms for fully nonlinear second order (time-independent) elliptic PDEs, the proposed research project intends to carry out a comprehensive and systematic study of numerical methods and algorithms for fully nonlinear second order (time-dependent) evolution PDEs. The objectives of the proposed research include (i) to develop the vanishing moment method and the moment solution theory for fully nonlinear second order evolution PDEs, (ii) to develop fully discrete Galerkin type numerical methods (e.g. finite element methods, mixed finite element methods, spectral and discontinuous Galerkin methods) for fully nonlinear second order evolution PDEs based on the vanishing moment methodology, (iii) to apply and/or to adapt the developed numerical methods to a number of emerging application problems which are governed by fully nonlinear second order PDEs, (iv) to develop computer codes for implementing the proposed numerical methods.
As numerical approximations of fully nonlinear second order evolution PDEs is an untouched sub-area within the numerical PDEs and those PDEs arise from many important applications in astrophysics, differential geometry, geostrophic fluid dynamics, image processing, kinetic theory, materials science, mass transportation, meteorology, and optimal control, the completion of the proposed research project is expected to have a profound impact on solving this class of PDEs and on providing the much needed capability and enabling tools for solving a range of important application problems which are governed by fully nonlinear second order PDEs. As a by-product, the moment solution theory is expected to give some insights to our understanding of the viscosity solution theory, and might be very likely to provide a logical and natural generalization and extension for the viscosity solution theory which is not natural and neither practical from the computational point of view. 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.
This research project studied numerical approximations of fully nonlinear second order evolution partial differential equations (PDEs). Such a class of differential equations has the strongest nonlinearity because they are nonlinear in the highest order derivative(s) of unknown functions and they arise from many scientific and engineering fields such as astrophysics, differential geometry, geostrophic fluid dynamics, image processing, materials science, optimal mass transportation, meteorology, and optimal control. They constitute the most difficult class of PDEs to analyze analytically and to approximate numerically. Building upon the PI's recent success on developing convergent and efficient numerical methods and algorithms for fully nonlinear second order (time-independent) elliptic PDEs, in this project the PI and his research team have developed a novel, nonstandard finite difference and discontinuous Galerkin framework for general fully nonlinear second order elliptic and parabolic partial differential equations. The research reveals the fundametal role that numerical moment plays in the framework and its connection with the vanishing moment method developed by the PI earlier. It also establishes a blueprint for developing efficient numerical methods for fully nonlinear second order PDEs in the near future. More than fifteen research papers (including a long review paper appeared in SIAM Review in 2013), which were partially supported by the grant, have been published in scientific journals during the grant period. The research results have also been reported in professional conferences and meetings. The grant provided valuable resources for partially supporting graduate research assistants for workng on their PhD degrees. Three PhD dissertations, which were partially supported by the grant, have been completed during the grant period. Two PhD students have successfully landed junior level academic jobs immediately upon their graduation. The numerical methods and algorithms developed in this project can be used to solve many application problems, which involve fully nonlinear second order partial differential equations, from scinetific and engineering fields such as differential geometry, antenna design, semi?-geostrophic flow, optimal mass transportation, astrophysics, fluid mechanisc, image processing, stochastical optimal contol, and mathematical finance.
