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The method relies on a conservative level set formulation for representing the fluid-fluid interface and uses adaptive mesh refinement on forests of octrees. An implicit time stepping with efficient block solvers for the incompressible Navier\u2013Stokes equations discretized with Taylor\u2013Hood and augmented Taylor\u2013Hood finite elements is presented. A matrix-free implementation is used that reduces the solution time for the Navier\u2013Stokes system by a factor of approximately three compared to the best matrix-based algorithms. 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