Stabilized finite element methods for solving the level set equation without reinitialization
In this paper we consider a level set equation, the solution of which (called level set function) is used to capture a moving interface denoted by Γ. We assume that this level set function is close to a signed distance function. For discretization of the linear hyperbolic level set equation we use standard polynomial finite element spaces with SUPG stabilization combined with a CrankNicolson time differencing scheme. Recently, in [Burmann, Comp. Methods Appl. Mech. Eng. 199, 2010] a discretization error bound for this discretization has been derived. The discretization induces an approximate interface, denoted by Γh. Using the discretization error bound, we derive bounds on the distance between Γ and its approximation Γh. From this we deduce a quantitative result on the mass conservation quality of the evolving approximate interface Γh. Results of numerical experiments are included which illustrate the theoretical error bounds.