  In numerical simulations of two-phase flows, the computation of the curvature
of the interface is a crucial ingredient. Using a finite element and level set
discretization, the discrete interface is typically the level set of a low
order polynomial, which often results in a poor approximation of the interface
curvature. We present an approach to curvature computation using an approximate
inversion of the $L^2$ projection operator from the Sobolev space $H^2$ or
$H^3$. For finite element computation of the approximate inverse, the resulting
higher order equation is reformulated as a system of second order equations.
Due to the Tikhonov regularization, the method is demonstrated to be stable
against discretization irregularities. Numerical examples are shown for
interior interfaces as well as interfaces intersecting the boundary of the
domain.
