The Goal of this thesis is to simulate the free flow and porous flow regions of a Proton Exchange Membrane Fuel Cell using a high-order Discontinuous Galerkin Method in Julia. Starting from an already existing DGM simulation of the Euler equations, firstly the compressible Navier Stokes Equations are to be simulated, and a penalty term will be implemented for the simulation of elliptic problems in general. Then, the time-stepping method for this problem will be chosen. An advection term will be added to the simulation to account for the flow of reactants of the oxygen reduction reaction. Furthermore, the possibility to incorporate this advection style equation into the NSE as source terms will be assessed. Lastly, the Darcy equations are implemented to account for the porous flow region, starting with an isotropic flow region, then an anisotropic flow region.