Eigenvalue problems for fully nonlinear elliptic partial differential equations with transport boundary conditions

Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods for nonlinear equations are extended to handle transport boundary conditions and eigenvalue problems. Finally, new techniques are designed to enable appropriate discretization of a large range of fully nonlinear three-dimensional equations

A Meshfree Generalized Finite Difference Method for Surface PDEs

In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinous surface properties (with large jumps) are also introduced, and a few practical applications are presented