  Unfitted finite element techniques are valuable tools in different
applications where the generation of body-fitted meshes is difficult. However,
these techniques are prone to severe ill conditioning problems that obstruct
the efficient use of iterative Krylov methods and, in consequence, hinders the
practical usage of unfitted methods for realistic large scale applications. In
this work, we present a technique that addresses such conditioning problems by
constructing enhanced finite element spaces based on a cell aggregation
technique. The presented method, called aggregated unfitted finite element
method, is easy to implement, and can be used, in contrast to previous works,
in Galerkin approximations of coercive problems with conforming Lagrangian
finite element spaces. The mathematical analysis of the new method states that
the condition number of the resulting linear system matrix scales as in
standard finite elements for body-fitted meshes, without being affected by
small cut cells, and that the method leads to the optimal finite element
convergence order. These theoretical results are confirmed with 2D and 3D
numerical experiments.
