An adaptive, structure-preserving numerical method for the rotating shallow water equations using hierarchical finite elements generated through subdivision

Recently, hierarchical finite element spaces were constructed using the subdivision algorithm
to generalize well-established concepts in isogeometric analysis to irregular meshes, for example of
the sphere. This talk illustrates the suitability of these function spaces for structure-preserving
simulations of the shallow water equations with local refinement. We focus on quantifying the
observed accuracy gains and the scaling of the compute time by comparison to established finite
elements without local refinement.
A key benefit of the hierarchical spaces is that they maintain a discrete de Rham complex
under local refinement and thus yield structure-preserving numerical methods. This property is
crucial for discretisations of atmospheric models as it allows us to locally increase the resolution
whenever necessary without changing characteristic features of the simulated systems, such as mass,
total energy or vorticity. Thus, the evolution of the discrete system mirrors the continuous system
faithfully and the simulation results are physically meaningful. We will present simulation results
that confirm these theoretical results.