During the first half of December, Felipe Sandoval and Cristian Inzunza successfully defended their undergraduate theses for obtaining the professional title of Mathematical Civil Engineer, both inspired by a completely mixed approach of the finite element method. With his thesis entitled: A Fully Mixed Finite Element Method for the Coupling of the Navier-Stokes and Darcy-Forchheimer Equations, Felipe was directed by the doctors Gabriel Gatica and Sergio Caucao, and his evaluation committee was additionally composed by Eligio Colmenares (UBB) and Ricardo Oyarzúa (UBB). In turn, the title of Cristian's thesis was: An Augmented Fully-Mixed Finite Element Method for a Coupled Flow-Transport Problem, and it was directed by Dr. Gatica. His evaluation committee was also formed by Jessika Camaño (UCSC) and Luis Gatica (UCSC). After successfully passing their undergraduate thesis, both young people aim to pursue doctoral studies either in Chile or abroad.

A novel area

The thesis made by Felipe Sandoval is a research through which he tried to find approximations of certain unknowns governing a problem that can be modeled with systems of differential equations, which depend on the nature of the physical phenomenon. "In this case, a fluid that interacts with a porous medium is modeled. We use the Navier-Stokes and Darcy-Forchheimer equations for the free fluid and the porous medium, respectively. It should be noted that mixed finite element theory was used to achieve the objective. The main applications are petroleum extraction, industrial leaks, the movement of blood in tumors, among others", explains the new graduate. This is a fairly novel area since, although in recent decades more emphasis has been placed on coupled problems, there is only a few previous works that deal with the same model, one of them precisely co-authored by Felipe's advisors. For him "my main motivation is to be able to contribute with the research of the Center for Mathematical Engineering Research (CI²MA) and, of course, in a certain way, to continue in the direction of the work already done by my thesis guides. Now, the main contribution of my work consisted of the possibility of approximating other unknowns of pysical interest of the model, which enhance the applicability of this method ".

Along with thanking the support of their supervisors, and highlighting that the work done in this thesis gave rise to a Preprint , Felipe points out that "the main results obtained are the usual ones with respect to investigations of this type; that is, under the established parameters, it is possible to prove that the continuous and discrete formulations have unique solutions and that the discrete solution is also a good approximation of the continuous one. Numerical essays illustrating the good performance of the method were also provided ".

A fully-mixed approach

A numerical analysis for the coupling of the Stokes equations with a transport problem, which describes the transport of a species by a fluid and involves non-linear functions, was the subject of Cristian Inzunza's thesis. In his research, both the Stokes and the transport equations were analyzed by using a mixed variational formulation. Although the theme itself is already quite known, in this case the main novelty is the fact that it is addressed with a completely mixed approach, which expands the variety of methods that can be employed to solve this type of problem. "In one of the courses I took with Professor Gatica, in fact the first one, we studied several problems and analyzed how to solve them using mixed finite elements and I wondered if I would be able to develop something similar to that. When I asked the professor for a thesis topic, he said that the fluid-coupled transport problem had been studied by making a mixed formulation in the fluid and primal in transportation, so it would be interesting to analyze what happened if a completely mixed approach was taken", he says. As results of the research, sufficient conditions were obtained to ensure the existence and uniqueness of solution of the continuous formulation, and the existence of solution of the discrete scheme. "In addition to obtaining approximations of the exact solutions, and thanks to the approach employed, the resulting method allows more freedom for choosing the finite element subspaces. Numerical essays confirming the theoretical results, were also provided", he says.

Cristian, who also remarks that his thesis gave rise to a Preprint, appreciates the disposition and confidence of Dr. Gatica that made possible his academic achievements and points out that "the next step is to develop the a posteriori errr analysis for the problem. On the other hand, we also want to adapt the approach from the thesis to the case of sedimentation-consolidation models ".