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

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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.

Original languageBritish English
Pages (from-to)138-145
Number of pages8
JournalApplied Mathematics Letters
Volume69
DOIs
StatePublished - 1 Jul 2017

Keywords

  • Finite element method
  • Navier–Stokes flow
  • Newton
  • Nonlinear problem
  • Surface tension
  • Third-order convergence

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