A Foreword to the Special Issue in Honor of Professor Bernardo Cockburn on His 60th Birthday: A Life Time of Discontinuous Schemings
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We present this special issue of the Journal of Scientific Computing to celebrate Bernardo Cockburn’s sixtieth birthday. The theme of this issue is discontinuous Galerkin methods, a hallmark of Bernardo’s distinguished professional career. This foreword provides an informal but rigorous account of what enabled Bernardo’s achievements, based on the concluding presentation he gave at the the IMA workshop “Recent Advances and Challenges in Discontinuous Galerkin Methods and Related Approaches” on July 1, 2017 which was widely deemed as the best lecture of his career so far.
With great enthusiasm, we are proud to present this special issue of the Journal of Scientific Computing in honor of Bernardo Cockburn on the occasion of his sixtieth birthday. It presents 24 selected papers submitted by the participants of the special workshop entitled “Recent Advances and Challenges in Discontinuous Galerkin Methods and Related Approaches” taking place at the Institute of Mathematics and its Applications (IMA) from June 29 to July 1, of 2017. The workshop aimed to bring together researchers at all levels and career stages, providing an opportunity to share and discuss recent progress in both theoretical and computational aspects of discontinuous Galerkin (DG) and related approaches. The list of attendees includes 111 participants from 15 countries/regions. The number and diversity of the participants, and the warm and supportive atmosphere of the community at display during the workshop are testaments to the high esteem in which Bernardo is held in the international scientific computing community. It is impossible to adequately cover the extent of recent advances in theory and computation of DG and related approaches in a single issue. This issue only intends to present a fragment of the timely research topics and noteworthy new developments in the area.
A central figure in this field, Bernardo received his B.S. and M.S. degrees from the Universidad Nacional de Ingeniería, his 3rd. Cycle Doctorat from Paris VI–IX, and his Ph.D. degree from the University of Chicago, all in Mathematics or Applied Mathematics. After a year as a postdoctoral fellow at IMA, he joined the School of Mathematics at the University of Minnesota in 1987. Having spent his entire career there, he currently holds a McKnight University Professorship. Bernardo’s work in the past three decades has made foundational impact in the field of discontinuous Galerkin methods, in particular, and in Scientific Computing and Numerical Analysis in general. An invited speaker at the International Congress of Mathematicians in 2010, Bernardo has been listed as a Highly Cited Author in Mathematics by the ISI Web of Knowledge. In addition to his unique contributions to the field encompassing both theoretical analysis and applications, Bernardo has also had a tremendous impact in the scientific community by mentoring and supporting junior researchers. These include his Ph.D. students, postdocs, visiting scholars, and young colleagues from all over the world.
2 The Academica Vitae Complex
So, a workshop like this is an ideal opportunity to experience a sense of belonging to a common undertaking as well as for expressing gratitude to the many people we met in this adventure.
2.1 A Sense of Belonging
Julio Ruiz-Claseyssen introduced him to the world of dynamical systems, and in particular to the work of his own adviser J. Hale. Guy Chavent introduced him to the application of the Discontinuous Galerkin method to oil recovery simulation and so, to the pioneering work of W.H. Reed and T.R. Hill, and that of P. Lesaint, and P. A. Raviart, as well as to the work on numerical methods for nonlinear hyperbolic conservation laws of A. Harten, P. Lax, B. van Leer, of B. Engquist and S. Osher, and to that of S. K. Godunov, S. N. Kružkov, and N. N. Kuznetsov. Finally, Jim Douglas, Jr., introduced him to the world of the finite element method and in particular to the work of J. H. Bramble, A. H. Schatz, J. A. Nitsche, and V. Thomée. Bernardo feels honored to have participated in the effort of trying to advance the knowledge in these areas.
Pictured in Figs. 2 and 3 are those Bernardo wanted to express sincere gratitude to for the success of this workshop. In particular, he appreciates the work by the workshop organizers Nilima Nigam (of Simon Fraser University), Bo Dong, Yanlai Chen (both of the University of Massachusetts Dartmouth), Wujun Zhang (of Rutgers University), and Ke Shi (of Old Dominion University). The workshop would not have been possible without the diligent support by the staff members of the IMA Samuel Richter, Georgia Kroll, Rebecca Malkovich, Chad Sullivan, IMA Assistant Director Katherine Dowd, and members of the School of Mathematics at the University of Minnesota Bonny Fleming and Harry Singh.
Bernardo thanked Julio Ruiz-Claeyssen for his dilligent and constant nurturing during his undergraduate times which included contacting J.-L. Lions (on the occasion of the Fourth Latin-American School of Mathematics in 1978) to secure his eventual working with Guy Chavent in France. He also thanked Víctor Sánchez-Moya, for supporting his internship in Petro-Perú, the Peruvian national oil company, and for inviting Guy Chavent to give a course on DG methods for oil simulation there. Finally, he thanked Jim Douglas, Jr., for his support during (and after) his Ph.D. work.
Bernardo recalled the joy of working with all his advisees during the years. He thanked his 26 doctoral graduates and 19 postdoctoral scholars for trusting him in such crucial moments of their academic life. Bernardo acknowledged the gracious support to all postdocs by institutions like the IMA, the Mathematical Sciences Postdoctoral Research Fellowships at NSF, the School of Mathematics, and numerous foreign governments. The fact that they all chose to work with Bernardo exemplifies the appeal of his research and his welcoming personality. Finally, Bernardo was grateful to all his collaborators for having “played” with him and to all the participants for their presence in the workshop.
3 Algorithm: The Tools of the Craft
Treasured as the second group are the “Rational thinking tools of the Enlightenment” (Figs. 5, 6, 7, 8). The best of them, as promptly pointed out by Bernardo, is to work with great collaborators – especially if they can write all the papers for you. Bernardo’s next crucial tool here is to stay focused in your direction avoiding discouragement. One fond story that Bernardo recalled is the review he received of his very first NSF proposal in 1988, not funded, on devising DG methods for nonlinear conservation laws. The reviewer concluded that “I think it is doubtful that important results will follow from this line of research”. In contrast, the world witnessed the explosive development of this exact line of research since then (Fig. 6).
As to the now often-used metric, the number of citations of an article, Bernardo jokingly pointed to his top-cited article (Fig. 7) which, by that account, received more than a thousand citations at the time but was read 0 times. He suggested to always take the number of citations with a grain of salt.
4 Numerical Results
It is the guest editors’ hope that this special issue will provide its readers an interesting glimpse into recent progress in the theory and computation of discontinuous Galerkin and related approaches. They would like to thank all the participants of the workshop, in particular the authors who submitted their papers to this issue, as well as the reviewers for a rigorous review process and timely and constructive comments. The role of IMA staff members Samuel Richter, Georgia Kroll, Rebecca Malkovich, Chad Sullivan, Assistant Director Katherine Dowd, and members of the School of Mathematics at the University of Minnesota Bonny Fleming and Harry Singh in the organization of the workshop can never be overstated, and their help is gratefully acknowledged. Last but not least, we express our sincere gratitude to Bernardo for the enlightening talk that inspired this article.