This thesis investigates the appliance of the finite element method to long wave propagation in channels. Weak formulations of potential- and velocity based Boussinesq equations were derived, with the normal dependent no-flux boundary condition imposed weakly, using Nitsche's method. Utilizing FEniCS, velocity based equations on constant depth has been thoroughly tested and show promising results when compared with the proven finite difference tsunami model, GloBouss. The FEM-solver's results in a sharp L-bend for different widths are in accordance to results in earlier papers and GloBouss. In a symmetric fork with branches at 45o the FEM-solver produces results that are reminiscent to other investigations of branched channels. Noticeable differences are observed contra results from GloBouss. Variable depth has been implemented in the FEM-solver, but not fully verified. Same applies to the potential based solver. Using the finite element method for solving the Boussinesq equations in complex geometries shows good promise, but need improvements to be efficient. Noticeable spikes are observed when solving for velocity, while the potential based solver seems to not have this non-physical behaviour.