Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow

In this letter, we present a computational framework based on the use of the Newton and level set methods and tailored for the modeling of bubbles with surface tension in a surrounding Newtonian fluid. We describe a fully implicit and monolithic finite element method that maintains stability for significantly larger time steps compared to the usual explicit method and features substantial computational savings. A suitable transformation avoids the introduction of an additional mixed variable in the variational problem. An exact tangent problem is derived and the nonlinear problem is solved by a quadratically convergent Newton method. In addition, we consider a generalization to the multidimensional case of the Kou's and McDougall's methods, resulting in a faster convergence. The method is benchmarked against known results with the aim of illustrating its accuracy and robustness.