His scientific research begins in the 90’s with the mathematical analysis of partial differential equations (PDEs) related to fluid mechanics, getting important new results for Navier-Stokes with variable density, non-Newtonian fluids, Primitive equations in oceanography, models for liquids crystals, etc.
In the 2000s, he also studies the numerical analysis of PDEs. As results, he designs new effective numerical schemes preserving properties of the continuous problem, obtaining moreover convergence, error estimates, etc. This research is complemented by numerical simulations using computer codes, which allows a rather complete study of PDE problems: mathematical and numerical analysis and scientific programming.
In the 2010s, he studies (analytical and numerically) phase field models widely used for modeling phase transitions (solid, liquid, gas and intermediate phases), and for the theory of mixtures or multi-components (immiscible fluids, vesicle-fluids interactions, etc.). This research is applied nowdays by a great quantity of research international group.
In the last years, he studies (analytical and numerically) PDE problems modeling Biological processes such as solid tumor growth, interactions living organisms vs chemical signal, etc., with special emphasis in the modelization of some real life situations and the solution of optimal control problems subject to these biological models. This research is allowing the modelization of some real life medical problems in collaboration with some I+D+i centers as IBIS (Institute of Biomedicine of Seville) and INiBICA (The Biomedical Research and Innovation Institute of Cadiz).
He has been the Principal Researcher in several projets financed by Spain governement (4 in the last 10 years). Principal researcher ofa Project financed by IEA-International Emerging Actions, in cooperation with Université de Nantes (France).
Its main international collaborators are: USA (G.Tierra, Texas), Germany (G. Grun), Brasil (B.Calsavara), Chile (E.Mallea-Zepeda), Colombia (E.J.Villamizar-Roa, D.Rueda-Gómez), etc.
He has been the Chair of two international conferences in 2019, sattelite events of ICIAM2019
Main research results
- P.Azerad, F Guillén-González. Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics. Siam J. Math. Anal. Vol. 33, 4 (2001), 847-859. Cites: 49 Scopus, 46 WoS.
Position 95/261 (2019 JCR Mathematics, Appplied).
Geophysical fluids all exhibit a common feature: their aspect ratio (depth vs horizontal width) is very small. By taking the aspect ratio goes to zero leads (at least formally) to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the Primitivie Equations.
In this paper, we prove rigourously this convergence process by means of anisotropic estimates and a new time-compactness criterium. These results have produced a high impact, because have been used after for the justification of several geophysical models for many international research groups.
- F Guillén-González, G Tierra. Numerical Methods for Solving the Cahn-Hilliard Equation and Its Applicability to Related Energy-Based Models. Archives of Computational Methods in Engineering, 22(1) (2015) 269-289. Cites: 49 Scopus, 45 WoS.
Position 1/101 (2017 JCR Mathematics, interdisciplinary applications).
Some numerical methods are presented to approximate the Cahn-Hilliard equation, which is the paradigm of conserved PDE model in phase transitions. The properties of each approach are compared to determine which one is better depending on what aspects are crucial to examinate: time accuracy, energy-stability, unique solvability and linear vs nonlinear schemes (considering the convergence of iterative methods towards the nonlinear schemes). The constraints arising on the physical and discrete parameters are even considered, detecting connections of the Cahn-Hilliard equation with other physically motivated systems. The point of view of this paper has been used in the last years for a great quantity of international research groups focused in the improvement of the numerical approximation for many different PDE models.
