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

Aymen Laadhari
Applied Mathematics Letters
Vol. 69, pp. 138--145, July 2017, in press

Abstract

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.


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@Article{eth_biwi_01359,
  author = {Aymen Laadhari},
  title = {Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow},
  journal = {Applied Mathematics Letters},
  year = {2017},
  month = {July},
  pages = {138--145},
  volume = {69},
  number = {},
  keywords = {Nonlinear problem; Super-convergence; McDougall’s method; Surface tension; Finite element method},
  note = {in press}
}