Numerical Methods and Algorithms for Fully Nonlinear Second Order Evolution Equations with Applications

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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.