Paulo Zúñiga is currently conducting a three-month research stay at the University of Waterloo (Ontario, Canada). He is a graduate student of the PhD program in Applied Sciences with minor in Mathematical Engineering of the Department of Mathematical Engineering and member of the Center for Restearch in Mathematical Engineering, CI²MA, Universidad de Concepción (UdeC). This research stay is financed by funds from Conicyt scholarship for Ph.D studies. 

Paulo obtained a Master degree in Mathematics from Universidad del Bío-Bío (UBB) in 2015 and currently he is working on his PhD thesis in the context of his Doctorate studies at UdeC (High-Order Mixed Methods in Continuum Mechanics) is directed by the CI²MA researchers Ricardo Oyarzúa (UBB) and  (UdeC). At Waterloo, Paulo explains, he is “collaborating with Dr. Sander Rhebergen, assistant professor of the Department of Applied Mathematics”, and regarding the objectives of his visit, the doctoral student explains that “the MacKenzie model is derived in terms of the Stokes equations for the earth's mantle and Darcy's Law for magma in the porous medium given by the earth's mantle, where the link between the two phases is the fraction of volume occupied by the magma or porosity; all parameters of the model evolve with it. When the porosity is zero and certain assumptions are satisfied, the system of equations is reduced to the problem of Stokes for incompressible flows in the earth's mantle. When the porosity is greater than zero, there are two phases: magma (fluid) and mantle of the earth (solid matrix). The goal of my research stay is to study the general case that describes the dynamics of the mantle/magma, when the porosity is zero or close to zero, which is a feature in most of the domains, because only in small regions the porosity is greater to 0”.

Paulo highlights the relevance of his stay in Canada in scientific terms: “Professor Rhebergen already has several work published about simpler versions of the equations with constant parameters, and more general cases where the null porosity is not taken into account in the analysis, evidencing the complications that can appear in the numerical approximations of the model when the porosity goes to zero”. This research visit, funded by the Scholarship Program for National Ph.D. Studies of Conicyt, was settled by Manuel Solano, in the context of his participation in the congress Recent Advances and Challenges in Discontinuous Galerkin Methods and Related Approaches, (2017, U of Minnesota, Minneapolis, United States) where he discussed collaboration projects with colleagues and friends that he had met years ago during his doctorate studies at the same American university. “Professor Rhebergen was among them” Paulo explains and adds that after that, they agreed to “work on MacKenzie's equations that model biphasic flows of partially molten regions of the earth's mantle, that is, the continuous dual mixing of magma and mantle of the earth. Geoscientists are interested on these regions because of their role in tectonic plates, volcanic activity, and the evolution of the earth”.

During his stay in Canada, Paulo will also participate in a session of talks about projects and publications that Sander Rhebergen regularly organizes, where each of his students present their progress. “My presentation will take place on March 23 and will be about the subject that I studied in my Master Thesis: Analysis of a finite element method for the Boussinesq problem with temperature-dependent parameters. I chose this topic because it is closer to the work that I have seen in Professor Rhebergen's group. It is also closely related to the MacKenzie problem that I am studying here. In addition, I will attend to the SIAM meeting on Applied Mathematics: SIAM Great Lakes Section Annual Meeting, which will take place on April 21 at Wayne State University, Detroit, USA. Detroit is relatively close to Waterloo, so I agreed to travel together with the graduate students of Professor Rhebergen. I get along very well with them”, he emphasizes, highlighting the kindness with which his colleagues, professors and staff have received him at the U. of Waterloo. “I’m very excited and enjoying a lot this internship because it slightly different to what I was working (high order numerical approximations for some PDEs on general domains), but at the same time, it remains in line of my research project. It is also a great challenge, mathematically speaking, that is just beginning and hope I can succeed”, Paulo says.