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An Unfitted Finite Element Method for Two-Phase Stokes Problems with Slip Between Phases

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Abstract

We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted generalized Taylor–Hood finite element pair \({\mathbf {P}}_{k+1}-P_k\), \(k\ge 1\), we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numerical results in two and three dimensions to corroborate the theoretical findings and demonstrate the robustness of our approach with respect to the contrast in viscosity, slip coefficient value, and position of the interface relative to the fixed computational mesh.

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Availability of data and materials

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We are grateful to Dr. Christoph Lehrenfeld for providing us with the ngsxfem script for two-phase Stokes with no slip.

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  1. Department of Mathematics, University of Houston, 3551 Cullen Blvd, Houston, TX, 77204, USA

    Maxim Olshanskii, Annalisa Quaini & Qi Sun

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  1. Maxim Olshanskii
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Correspondence to Annalisa Quaini.

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This work was partially supported by US National Science Foundation (NSF) through Grant DMS-1953535. M.O. also acknowledges the support from NSF through DMS-2011444. A.Q. also acknowledges the support from NSF through DMS-1620384.

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Olshanskii, M., Quaini, A. & Sun, Q. An Unfitted Finite Element Method for Two-Phase Stokes Problems with Slip Between Phases. J Sci Comput 89, 41 (2021). https://doi.org/10.1007/s10915-021-01658-x

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