Mixed Finite Element Methods for Time-Dependent Wave Propagation Problem: Elastodynamics and Elastoacustics is the name of the thesis developed by Carlos García whose successful defense earned him to degree of Doctor in Applied Sciences with a major in Mathematical Engineering granted by the Universidad de Concepción (UdeC).

In performing his research, García worked under the direction of the Center for Research in Mathematical Engineering (CI²MA) director Gabriel Gatica and the co-direction of the academic of the Department of Mathematics of Universidad de Oviedo, Salim Meddahi. "Carlos has shown a lot of enthusiasm and a lot of thoughtfulness in his research work. In addition, the excellent training he has received has allowed him to quickly learn the necessary techniques to advance on the work of his thesis", Meddahi said about his experience working with García, after the research stay the UdeC student did in 2014 at Universidad de Oviedo. In addition to the advisors Gatica and Meddahi, the Examination Committee was composed of professors Andrés Ávila (Universidad de la Frontera, Temuco) and Ricardo Oyarzua (Universidad del Bío-Bío, Concepción), who is also an associate researcher of CI²MA.

Carlos is a Mathematical Civil Engineer of the UdeC and when facing his postgraduate studies decided to focus his research in Numerical Analysis of Partial Differential Equations (PDE). “It was natural for me to enter this area, because at CI²MA there are many researchers who are very strong in it. I have always been interested in working with evolutionary PDE, which means that they involve time and describe the behavior of a system and how it evolves over time. It is not a common thing to research about here, in contrast to stationary issues. That is why professors Gatica and Meddahi proposed to me to work on a study related to elastodynamics. In general, it is an area where there is a broader range of action and a longer path to travel”, he explains.

“In the thesis, we worked with Mixed Finite Element Methods for wave propagation problems, which are evolutionary; in particular, two: elastodynamics and elastoacoustics. We study the oscillations of a vibrating object through the construction of PDE containing terms with derivatives with respect to time. So I was interested in appraising some numerical methods that would allow us to approximate the solution”, Carlos said. He also said that in the immediate future he is preparing his presentation at the Santiago Numérico III Congress that will be held in a few weeks time. In addition, he plans to perform postdoctoral work abroad, which he is preparing his application for. “I look forward to doing a postdoc with Professor Jay Gopalakrishnan at Portland State University (Oregon, USA). I contacted him thanks to Professor Gatica, after the research visit I did in 2015 at the University of Oviedo”, García explains. Among other immediate goals, Carlos sustains that “within this year, I hope to produce a new publication, either with an alternative solution to the problem of elastodynamics already described in my thesis, or an a posteriori error analysis for it” states the new Ph.D, who already has 4 papers: one published, two accepted, and one sent.

As for the projection of the results of Garcia's work, Professors Gatica and Meddahi emphasized that “Carlos could complement the work of his thesis with an a posteriori analysis of the method of mixed finite elements for elastodynamics. Also, it would also be interesting to obtain a scheme with strong symmetry for the stress tensor and thus to eliminate the auxiliary variable called rotation. Finally, we suggested that he tried to think of an iterative method that simplifies the numerical solution of the problem of fluid-solid interaction”, the academicians said.