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In this work, we consider unfitted finite element methods for the numericalapproximation of the Stokes problem. It is well-known that this kind of methodslead to arbitrarily ill-conditioned systems. In order to solve this issue, weconsider the recently proposed aggregated finite element method, originallymotivated for coercive problems. However, the well-posedness of the Stokesproblem is far more subtle and relies on a discrete inf-sup condition. Weconsider mixed finite element methods that satisfy the discrete version of theinf-sup condition for body-fitted meshes, and analyze how the discrete inf-supis affected when considering the unfitted case. We propose different aggregatedmixed finite element spaces combined with simple stabilization terms, which caninclude pressure jumps and/or cell residuals, to fix the potential deficienciesof the aggregated inf-sup. We carry out a complete numerical analysis, whichincludes stability, optimal a priori error estimates, and condition numberbounds that are not affected by the small cut cell problem. For the sake ofconciseness, we have restricted the analysis to hexahedral meshes anddiscontinuous pressure spaces. A thorough numerical experimentation bears outthe numerical analysis. The aggregated mixed finite element method isultimately applied to two problems with non-trivial geometries.

Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem