An optimization-based approach to enforcing mass conservation in level set methods

This paper presents a new conservative level set method for numerical simulation of evolving interfaces. A PDE-constrained optimization problem is formulated and solved in an iterative fashion. The proposed optimal control procedure constrains the level set function to satisfy a conservation law for the corresponding Heaviside function. The target value of the state variable is defined as the solution to the standard level set transport equation. The gradient of the control variable corrects the convective flux in the nonlinear state equation so as to enforce mass conservation while minimizing deviations from the target state. A relaxation term is added when it comes to the design of an iterative solver for the nonlinear system. The potential of the optimization-based approach is illustrated by two numerical examples.