Fully implicit finite element method for the modeling of free surface flows with surface tension effect

A new approach based on the use of the Newton and level set methods allows to follow the motion of interfaces with surface tension immersed in an incompressible Newtonian fluid. Our method features the use of a high-order fully implicit time integration scheme that circumvents the stability issues related to the explicit discretization of the capillary force when capillary effects dominate. A strategy based on a consistent Newton–Raphson linearization is introduced, and performances are enhanced by using an exact Newton variant that guarantees a third-order convergence behavior without requiring second-order derivatives. The problem is approximated by mixed finite elements, while the anisotropic adaptive mesh refinements enable us to increase the computational accuracy. Numerical investigations of the convergence properties and comparisons with benchmark results provide evidence regarding the efficacy of the methodology. The robustness of the method is tested with respect to the standard explicit method, and stability is maintained for significantly larger time steps compared with those allowed by the stability condition.