During his time as a PhD student Nelson Faustino worked on the fundamentals of discrete function theory and its applications to numerical solution of partial differential equations. While there already exist some results in this direction, such results were mostly obtained by “brute force calculations”, not by a well-established theory.
In order to create such a theory, Faustino systematised the abstract approach to continuous function theory by F. Sommen, V. Kisil and others by showing it to be a combination of umbral calculus and differential geometry. This allowed the construction of intertwining operators between continuous function theory and certain discrete versions. With such operators one can lift known results in the continuous case to the discrete case, which provides the construction of basis polynomials, Stokes formula, Borel-Pompeiu formula, Cauchy integral formula, etc. This rather elegant way allows us to avoid an the cumbersome work with obscure analogies. On the other hand, the resulting construction based on discrete differential forms and discrete integration in terms of barycenter coordinates provides also an early correspondence between the theory of finite differences and the theory of finite elements.
Furthermore, because of the interest in his research, Faustino got four invitations for research stays in Tampere University of Technology (Finland), Konrad-Zuse-Institut f¨ur Informationstechnik Berlin (Germany), Bauhaus-University Weimar (Germany), and the University of Leeds (UK).
During his time as a FCT-research fellow Faustino authored 9 publications and preprints of which at least 6 have appeared or are accepted for publication in international journals and in two of them he is the sole author. Faustino gave 20 talks, 11 of them in International Conferences and Workshops and two of them during the research stays in Bauhaus-University Weimar (Germany) and the University of Leeds (UK).
His approach and his (first) results already started to received a lot of interest during his talk at the CMFT-conference in 2005 in Joensuu (Finland) at the beginning of his Ph.D work. The presented results contributed to the article Difference potentials for the Navier-Stokes equations in unbounded domains (joint work with K. Guerlebeck & Angela Hommel, Bauhaus-University Weimar and U. K¨ahler, University of Aveiro) published in the Journal of Difference Equations and Applications. It should be stressed that this is a UK Mathematical Journal with a considerable impact factor.
After his participation in the International Congress of Mathematicians in 2006, Madrid (Spain) where Faustino also delivered a talk, he made some contributions to the development of the theory of discrete Dirac operators and the numerical solution of timedependent non-linear Schr¨odinger equation.
We would like to point out that F. Sommen is a prolific researcher in Clifford Analysis. He was author of 3 books in the field and his list of publications contains more than 100 papers. We would like also to point out that the second article were published in a mathematical journal with a considerable impact factor. The editorial board of the journal include well-known names from the Mathematical community like Ivo Babuska (University of Texas at Austin, USA), Franco Brezzi (Universita di Pavia, Italy), P. G. Ciarlet (Universit´e Pierre et Marie Curie, Paris, France), Endre Suli (University of Oxford, UK), Roger Temam (Indiana University, USA).
Besides the above topic Faustino got interested in the Huang-Hilbert transform and analytic signals, he constructed and implemented working algorithms in the case of higher
dimensions (Huang’s original approach being one-dimensional). This is joint work (under construction) with P.Cerejeiras, U. K¨ahler (University of Aveiro, Portugal) and G. Teschke (Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin). We would like to remark that G. Teschke is an active researcher with Scientific and Industrial Projects Funded by Industries and Public Grants such as DFG, AIF, BMBF, DAAD. His scientific partners include well-know names like Ingrid Daubechies (Princeton University, USA), Stephan Dahlke (University of Marburg, Germany) and Luminita Vese (University of California Los Angeles, IPAM).
Nelson Faustino’s CV shows that he is a talented mathematical researcher as well as good in finding concrete computational solutions for specific mathematical problems.
His research interests include very distinct topics including Clifford Analysis, numerical analysis, wavelet analysis, umbral calculus, quantum mechanics and signal processing.
Faustino also finished his Ph.D at March 26, 2009.