In this thesis report, we investigate the difference in using physics-compatible elements and standard elements in the context of fluid flow through porous media. For this, we compare the H(div)-conforming Raviart-Thomas elements to Lagrange elements. Our model involves a channel flow over a porous media, resulting in a fluid and a porous subdomain. We distinguish between three approaches, as based on a research paper. The first is the one-domain approach called PE, which sets one equation on the entire domain. The other two approaches are two-domain approaches, which use different equations in the two subdomains, with interface conditions to couple them. The first of these is NSD, which couples Navier-Stokes with the Darcy equation. The second is NSF, which does the same but also incorporates the Forchheimer term, which models inertial effects.

In general, we were not able to make a completely successful model, as the PE case did not match the paper, and the results found when using Raviart-Thomas elements had some unexplained anomalies. We were also unable to carry out an accurate error analysis. We were able to see that Raviart-Thomas produced an exactly divergence-free solution, as expected from the theory, at the cost of more computing power and time needed.

This report has the main aim of comparing the Mixed Finite Element Method to the standard Finite Element Method. The other aim is to let the reader understand what these methods entail. The latter is done by first journeying through the theory behind the FEM. It is first explored in one dimension to keep the setting simple. Next, the Mixed FEM is explored. A new way of approximating the gradient from the found solution is constructed, using the basis of Finite Differences and the ideas from the Finite Volume Method. Following the theory is the implementation of the mentioned methods and the analysis of theresults. It yields that, when looking at the L2-norm, the classic FEM has a convergence order of 2, comparable to that of similar numerical methods. The Mixed FEM seemed to converge with an order of 4 in the same norm. Our constructed method only had a measly first order convergence, implying much greater accuracy of the Mixed FEM.