Title：Six Corelated Ways of Moving Mesh Adaptations for Nonlinear Kawarada Problems
It was during the last Banff International Research Station (BIRS) Workshop on Adaptive Numerical Methods for Partial Differential Equations with Applications, the topics of different adaptive finite difference procedures were seriously revisited, reevaluated and reinvested. Among various kinds of singular partial differential equation problems, Kawarada problems are particularly attractive to the participants due to their important theoretical and application features.
In a traditional moving mesh approach, mesh adaptations are often configurated based on an equidistribution principle. In such a case, a new mesh is acquired via a monitor function that is equidistributed in some sense. Typical choices of such monitor functions involve the solution or one of its many derivatives. The strategy has been proven to be effective and easy-to-realize in multi-physical applications. However, identifications of optical core monitoring components are proven to be extremely difficult. To this end, in this talk, we consider six different designs of monitoring functions targeting at a highly vibrate nonlinear partial differential equation problem that exhibits both quenching-type and degeneracy singularities. While the first a few monitoring designs to be discussed are within the so-called direct regime, the rest belong to a newer category of the indirect type, which requires the priori-knowledge of certain important solution features or characteristics. Computational experiments will be presented to illustrate our research and conclusions. Continuing collaborations in the field with SEU colleagues and students will also be aimed.
Dr. Sheng received his BS and MS in Mathematics from Nanjing University in 1982, 1985,respectively. He then obtained a Ph.D. degree from the University of Cambridge under the supervision of Professor Arieh Iserles. After his postdoctoral research with Professor Frank T. Smith, FRS, in University College London, he joined National University of Singapore in 1990. Since then, Dr. Sheng was on faculty of several major universities till his joining Baylor University, which is one of known research institutions and the second biggest privateuniversity in the United States. He has been associated with the Mathematics Department andCenter for Astrophysics, Space Physics and Engineering Research (CASPER). See moredetails from Dr. Sheng's website https://sites.baylor.edu/qin_sheng/ .
Prof. Sheng has been interested in adaptive and splitting computations for solvinglinear and nonlinear partial differential equations. He is also known for the Sheng-Suzuki theoremin numerical analysis. He has published over 110 refereed articles as well as several joint research monographs. He has been an Editor-in-Chief of the SCI journal, International Journal of Computer Mathematics by Taylor and Francis Group since 2010. He gives invited presentations,including keynote lectures, in international conferences each year. Dr. Sheng's projects have beensupported by several research agencies. He currently advises 2 doctoral students and 2postdoctoral research fellows. He also has several academic visitors from overseas institutions.