Computational Fluid Dynamics (CFD) offers numerous benefits, notably the ability to study flows that are challenging or costly to investigate using experiments. A central challenge in CFD lies in simulating fluid flow around complex geometries. Additionally, the governing equations follow conservation laws. This thesis aims to establish the foundation for constructing Immersed divergence-conforming finite element spaces that address both challenges. To tackle the first issue, an immersed method is considered. Instead of generating a mesh that conforms to the object where the flow occurs, the approach involves placing the object within a predefined mesh. However, this introduces new challenges, the most significant of which is the ill-conditioning of the associated linear system. The condition number depends on the location of the immersed object and can lead to an extremely large condition number. The second challenge is resolved by discretizing the problem in a way that preserves essential topological and homological structures at a discrete level, utilizing the principles of finite element exterior calculus. Within this thesis, a structure-preserving subcomplex is developed for the de Rham complex in 1D, and its accuracy is validated through numerical experimentation. Optimal convergence is achieved, the discrete inf-sup test is satisfied, and the resulting linear system's conditioning remains unaffected by the placement of the immersed object. However, the constructed structure-preserving subcomplex for the de Rham complex in 2D, known as the immersed divergence-conforming finite element spaces, does not exhibit convergence. For future research, I recommend focusing on simpler vertical and skewed cuts before exploring more intricate immersed shapes. These simpler cuts involve straightforward choices for edge basis functions/1-form basis functions that are relevant. Additionally, if the outcomes remain unsatisfactory, considering a global approach instead of a local approach could be worthwhile.

In this thesis the water motion and sediment dynamics are investigated in a periodically closed and opened estuary. The water motion in an estuary is mainly driven by the semi-diurnal tide with an period of 12h25m and river discharge. An example of such an estuary is the Ems-Dollard estuary. Recent observations show an increase in tidal range (height difference between high and low tide), suspended sediment concentration and the depletion of oxygen levels (consequently harming the ecosystem). A possible solution, periodically closing and opening the estuary, is investigated. The water motion in a periodically opened and closed estuary is described by the linearised cross-sectionally averaged equations which give the sea-surface elevations and tidal velocity when solved with the eigenfunction expansion method. It was found that the sea-surface elevations and tidal velocity for a periodically opened and closed estuary are again 12h25m periodic. For the sediment transport, when no overtide is considered the residual sediment transport is seaward directed if the estuary is closed at low water and landward directed if the estuary is closed at high water. The barrier location determines the magnitude of the residual sediment. When overtide is included in the forcing of the system no relation is found
between the direction of the residual sediment transport and the closing height and closing position. The location of the barrier and closing height both determine the magnitude of the residual sediment transport and direction. By introducing a barrier that periodically closes and opens we intended to achieve a seaward directed residual sediment transport in the Ems-Dollard estuary. The results suggest that this is not possible. Further research is needed with more
extensive models to confirm this. For future Research I recommend to extend the model to a two-dimensional model with the eigenfunction expansion method. Other possibilities may be to consider a spatial dependent erodible bed.