# Numerical diffusion in during advection of a free surface in an FE context

I am currently working on a project where a two-phase flow is considered. The phases are described using a level set approach and a signed distance function from the interface between the phases where positive values are liquid and negative values are gas.

I consider a Stoke's flow which gives me a velocity field $v$ and a pressure field $p$. Then I use $v$ to advect the level set using the advection equation

$\frac{\partial \phi}{\partial t} + v \cdot\nabla\phi=0$.

I am new to this type of problem but I think I have some basic understanding on the numerical (diffusion) problems arising from this equation due to discretization and I know that I need some kind of stabilization. My research concerns the modelling of two-phase flow in a porous material and the results needs to be quite accurate, hence the popular SUPG scheme does not quite cut it.

My question is simply, what schemes (for Finite Elements) exists, that gives a better solution than the SUPG scheme? And furthermore, is the any good literature on the topic that discuss, and preferably, compares any stabilization schemes?

• For pure advection such as this, you might consider Discontinuous Galerkin methods. They are FE methods designed to solve precisely this sort of problem and can give you straightforward means to achieve high orders of accuracy. – Tyler Olsen Mar 10 '17 at 14:54
• There is a wealth of information on solving the advection equation. One highly-regarded text is by Leveque amazon.com/dp/0521009243/ref=rdr_ext_tmb – Bill Greene Mar 10 '17 at 16:11
• Concerning the Discontinuous Galerkin methods: Since I have a free surface flow (with suface tension) I need something that is continuous over element boundaries in order to accurately estimate the surface tension. An alternative would of course be to recreate the surface, but I would prefer some CG method. From what I understand, the Galerkin/Least-squares method performs quite well, so that might be my scheme of choice. – Carl Mar 15 '17 at 6:34