In this talk, we present an Unfitted Space-Time Finite Element method for the scalar transport equation posed on moving domains. We consider the case of the domain boundary being transported by the same velocity field as the scalar concentration inside the physical domain. A standard continuous Galerkin Finite element space is considered on a fixed background mesh, as well as tensor product Space-Time elements, which can be discontinuous along time slice boundaries. For the computational geometry, we opt for a spatially second-order accurate approximation variant in the mathematical analysis. In particular, we establish stability in a problem-specific norm and prove a priori error bounds of high order. Numerical examples illustrate these theoretical findings.

Cut cells provide efficient meshing techniques based on a simple idea: The geometry of interest is placed in a cartesian (background) mesh. Intersections of the geometry with cartesian mesh elemetns are then removed, leading to so called cut cells while the rest of the background mesh stays untouched. While simple and efficient, the cut cells in the mesh can be of arbitrary size and shape. In the context of hyperbolic conservation laws, where explicit Runge-Kutta methods are very popular, stabilization is needed to achieve a reasonable time step size that is independent of the size of small cut cells in the mesh.
By now a variety of stabilization methods have been developed that successfully address this issue. However, the question of structure-preserving properties of the resulting schemes is still delicate: Even if the stabilization is applied to a structure-preserving base scheme, it still has to be carefully constructed to not lose any desired properties.
In this contribution we will consider the Domain of Dependence stabilization and discuss how it can be applied to an entropy-preserving DG scheme without destroying entropy preservation.

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.

We study a DG(1) time-discretisation of the heat equation on a moving domain in Eulerian coordinates. Despite the changing spatial domain between consecutive time intervals, the previously computed solution is used only where it is naturally defined and therefore requires no extension. The spatial discretisation is based on a cut finite element method (CutFEM).
To derive practical time-stepping schemes, the arising time integrals are approximated by Trapezoidal or Simpson’s rule.
A priori error estimates in space and time are derived for the fully discrete schemes, providing convergence results for the combined DG(1)-in-time and unfitted finite element discretisation.
Finally, numerical results in two spatial dimensions are presented.