Simulation Tools for Three Dimensional Fluid Flow with Electrostatic Forcing Term on Arbitrary Geometry Using Octree Grids

Partial differential equations on irregular domains are ubiquitous in science and engineering. One of the main challenges with their numerical approximations is how to represent the geometry and how to impose the boundary conditions at the domain's boundary. In addition, it is typical that such problems involve different length scales so that computations on uniform grids is inefficient at best and often computationally prohibitively expensive. In this dissertation, we present innovative solutions to address those challenges in the case of three different partial differential equations that are encountered in several classes of science and engineering problems. The first describes the numerical solution of the Poisson equation where jump conditions are enforced at irregular interfaces. This is the case for example of multiphase flows, where the pressure experiences a jump across the interface between two phases. The method we propose is the first approach to use Quadtree grids, a data structure proven to be optimal in terms of CPU and memory requirements. We then turn our focus to the numerical solution of the Poisson-Boltzmann equation with Neumann or Robin boundary conditions. This system describes at the continuum level the steady state electric potential in an ionic solution and its numerical simulation has been motivated by the recent thrust in research at the micro and nano scale. Our method is the first to consider those boundary conditions for the Poisson-Boltzmann equation with irregular interfaces and also the first to use Quadtree and Octree grid structures. Finally, we introduce a numerical method for solving the Navier-Stokes equation coupled with the Poisson-Boltzmann equation. This system of equations describes the flow in micro and nano channels induced by electrostatic forces. Overall, the numerical methods presented allow for an accurate description of the different length scales, while offering the ability to simulate three dimensional problems. The irregular interfaces are implicitly represented using the Level Set method and boundary conditions are enforced using a hybrid Finite Difference and Finite Volume approach. Adaptive Quadtree and Octree grids are shown to save significant computational time and memory. Accuracy is investigated throughout this dissertation.