Finite element methods for non-divergence form partial differential equations and the Hamilton-Jacobi-Bellman equation

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Many models in the sciences and engineering are solved approximately using computational methods, and it is necessary to theoretically justify the reliability of the computed approximations. In addition to providing justification of the numerical methods, the theoretical analysis often provides insight for the development of new methods with improved efficiency, accuracy, and viability. In this project, the investigator and a graduate student will construct, analyze and implement numerical methods for classes of linear and nonlinear partial differential equations arising in stochastic financial models, stochastic differential games, and other applications in finance and engineering. The overall aim of the project is to develop methods that can be implemented using current computational software and to derive explicit estimates of the approximate solutions. The main goal of this project is to propose robust finite element discretizations for a class of fully nonlinear Hamilton-Jacobi-Bellman (HJB) equations and to develop a comprehensive convergence theory. The project consists of two integrated components: (1) The development of finite element methods for second order elliptic equation in non-divergence form with non-smooth coefficients; the building-blocks of the HJB problem, (2) The construction, implementation and convergence analysis of practical finite element discretizations for the HJB problem. The work will broaden the mathematical theory of the finite element method to problems that have been relatively untouched in the numerical community. The success of this project will have broad impacts in the mathematical and computational applications of stochastic optimal control in finance and engineering. In addition, the impacts of this project are felt though the training of the next generation of computational scientists, course development, and dissemination to the mathematical and scientific computing communities.