Yen Ting Ng
November 23, 2010
1132 Harold Frank Hall
Frederic Gibou (chair)
Title: Boundary Treatments for Free Boundary Problems in complex geometries
Free boundary problems are ubiquitous in science and engineering and are notoriously difficult to solve numerically. The difficulty stems for a large part from the fact that complex boundary conditions for partial differential equations must be imposed on irregular shapes in two and three spatial dimensions. In particular, the Poisson equation is one of the building blocks in partial-differential-equation based modeling of physical phenomena and has countless applications in fluid dynamics, heat transfer, electrostatics, wave phenomena and a range of other important engineering problems. For example, the Poisson equation is integral to the projection method for solving the Navier-Stokes equations, and methods for solving the Poisson problem can be extended to the more general heat equation and Stefan-type problems. Many different approaches have been proposed for solving the Poisson problem subjected to different boundary conditions, the goal being a computationally efficient method that is extensible to three or more spatial dimensions and produces a high order of accuracy in the solutions and gradients. Towards this goal, we will discuss our work in 1) analyzing the requirements for producing second-order accurate solutions and gradients in an existing method, 2) a novel approach for imposing Neumann boundary conditions on irregular domains in the context of single-phase flows, and 3) a straightforward method for handling mixed Dirichlet-Neumann boundary conditions on irregular domains.