This thesis details two numerical methods for the solution of incompressible flow problems using the virtual node framework introduced in (Bedrossian, 2010). The first method is a novel discrete Hodge decomposition for velocity fields defined over irregular domains in two and three dimensions. This new decomposition leads to a sparse, 5-point stencil in 2D (7-point in 3D) at all nodes in the domain, even near the boundary. The corresponding linear system can be factored simply into a weighted product of the standard discrete divergence and gradient operators, is symmetric positive definite, and yields second order accurate pressures and first order velocities in the maximum norm (second order in the 1-norm).

The second method is an extension of the work in (Assenço, 2013), which simulates the Stokes equations in two dimensions, to a method that models the Navier-Stokes equations in two and three spatial dimensions. The extension to three dimensions is partially accomplished by a new approach to discretizing the multiplier term corresponding to the system jump conditions. This method works either on domains with interfacial discontinuities in material quantities such as density and viscosity, or on irregularly shaped domains with Dirichlet, Neumann, or slip boundary conditions. This method leads to a discrete, KKT system solving for velocities and pressures simultaneously, and yields second order accurate velocities in both time and space, and first order pressures.