A Tikhonov approach to level set curvature computation

TL;DR

This paper introduces a Tikhonov regularization-based method for more accurate and stable computation of interface curvature in two-phase flow simulations using level set and finite element discretizations.

Contribution

It proposes a novel approach to approximate the inverse of the $L^2$ projection operator for curvature computation, improving stability and accuracy over traditional methods.

Findings

Method is stable against discretization irregularities.

Numerical examples demonstrate improved curvature approximation.

Applicable to interfaces intersecting domain boundaries.

Abstract

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.