I am looking for easily to understand example of level set method used to track phases interfaces. I would like to solve it using FEM because my solution is based on the FEM solution of second Fick law. Initialization step is straight forward - it just distance of each node from the interface. But I have problem with second step - solution of the interface velocity equation:$$ \frac{\partial \phi }{\partial t}+F\left| \nabla \phi  \right|=0 $$If I understand correctly I need a weak formulation of that differential equation, where F is linear velocity of the interface. Solution will give me a new position of the interface. Is it correct? 
Example of level set method
finite element
TL;DR: Yes, $\phi=0$ always locates the interface even at later times. See below for more details:A level set function, $\phi(x,y,t)$, in two dimensions, describes a function whose zero-level set describes your interface. To be precise, a level-set function $\phi$ divides your domain $\Omega$ into three disjoint subdomains:$$\left\{\begin{array}{lcr}\Omega^- &:(x,y,t) \:&|\: & \phi(x,y,t) < 0 \\\Gamma   &:(x,y,t) \:&|\: & \phi(x,y,t) = 0 \\\Omega^+ &:(x,y,t) \:&|\: & \phi(x,y,t) > 0 \\\end{array}\right.$$Assuming that $\phi(x,y,t) = 0$ always denotes the interface $\Gamma$, it is clear that the the interface must satisfy:$$\frac{D \phi}{D t} = \frac{\partial \phi}{\partial t} + \mathbf{V} \cdot \nabla \phi = 0$$Furthermore, since $\phi(x,y,t) = 0$ on $ \Gamma$, this equation may be written to only involve the velocity in the normal direction $$\frac{\partial \phi}{\partial t} + v_n \frac{\partial \phi}{\partial n} = 0$$However, since $\Gamma$ is a level-set, we have the following relation for the unit normal:$$\hat{\mathbf{n}} = \frac{\nabla \phi}{\left|\nabla \phi\right|}$$Thus, the evolution equation simplifies to:$$\frac{\partial \phi}{\partial t} + v_n \left|\nabla \phi\right| = 0$$This is your version of the equation with $F = v_n$. It is sometimes referred to as motion in the normal direction. To summarize, the zero level-set of $\phi(x,y,t)$ always locates the interface when you solve the level-set equation (either in its original form or normal motion form). I suggest you consult the following two standard texts for more details and actual implementation:Sethian's bookOsher & Fedkiw's bookThe first one includes a chapter on weak formulation and FEM methods for the solution whereas the second is mainly involved with FDM methods.