When construting discretizations of complex shaped domains, a major bottelneck is still the mesh generation. Here cut-cell meshes can be attractive, as they eliviate major challenges of mesh generation. The idea is to create a beckground mesh without taking geometric features into account and then intersecting the individual cells with the actual geometry. While this technique is relatively simply, it comes at a price. We loose control over the mesh regularity and can get highly anisotropic and arbitrarily small cells. This loss of mesh regularity is either counteracted technically by again modify the mesh or numerically by different stabilization techniques.

The domain of dependence (DoD) stabilization was initially developed to overcome stability problems of discontinuous Gelerkin (DG) discretizations of hyperbolic conservation laws on cut-cell meshes, in particular restrictive time step sizes. It is based on a weak reconstruction of the domain of dependence of the discrete operator and by this allowing time step sizes that only depend on the mesh size of the background mesh. In this talk we show how the DoD method can also be used to construct new stabilization techniques cut-cell discretizations of elliptic problems and discuss similarities and differences to existing techniques, in particular the ghost-penalty method.