Past Event: Oden Institute Seminar

Analysis of the DG method applied to an elliptic problem with nonlinear Newton boundary conditions

Miloslav Feistauer, Professor, Charles University, Faculty of Mathematics and Physics Prague, Czech Republic

Abstract

The lecture will be devoted to the analysis of the discontinuous Galerkin method (DGM) for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. Problems of this type appear in the modeling of electrolysis of aluminium, radiation heat transfer problems or in nonlinear elasticity. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate DG solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the singular boundary points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and the nonlinearity in the boundary term. It also depends on the parameter defining the nonlinear behaviour of the Newton boundary condition. Important tools in the analysis are the results of A. Buffa and C. Ortner on compact embeddings of broken Sobolev spaces. Theoretical results are accompanied by numerical experiments showing nonstandard behaviour of the error in the L2-norm and the H1-seminorm. The results were obtained in cooperation with Filip Roskovec and Anna-Margarete Sandig. Bio Prof. Dr. Miloslav Feistauer, DrSc., Dr.h.c., professor of mathematics at Charles University in Prague, was born in Náchod into a family of teachers. From an early age he was interested in mathematics and physics, and also devoted himself to music. After eleven years at high school in 1960, wondering whether to study violin or mathematics, he decided to study at the Faculty of Mathematics and Physics, Charles University. There he has spent all his professional life. After graduating in Applied Mathematics in 1965, he joined the Department of Applied Mathematics. After three years he was appointed lecturer and in the following year received the degree of Doctor of Natural Sciences (RNDr.). In 1972 he defended the scientific degree of Candidate of Sciences (Ph.D.) and in 1982 took a habilitation in mathematics. However, since Prof. Feistauer had From Pokroky Mat. Fyz. Astronom. 58 (2013), 73–75. Translated by Miluša Bubeníková 489 never been politically active, his appointment to the position of associate professor was held up till 1988. In 1990 he was awarded the degree of Doctor of Sciences (DrSc.) and shortly thereafter in 1991 was appointed full professor of mathematics in the field of approximate and numerical methods. In this position he has worked up to now. In the period of 1986–1994 he worked at the Mathematical Institute of Charles University and in 1994 he gained the post of the head of the Department of Numerical Mathematics at Charles University in open competition. In this position he served till 2006. Professionally, Prof. Feistauer's scientific activity has been in the areas of partial differential equations, numerical and computational mathematics and mathematical methods in fluid dynamics. More than 200 papers, 3 monographs, more than 130 invited lectures at universities, more than 180 lectures at conferences (including 68 invited main or plenary lectures and 15 lectures in Oberwolfach), organizing conferences in the Czech Republic and abroad (e.g., member of the Program Committee of the series of the conferences ENUMATH). Teaching activities of Prof. Feistauer are in accord with his successful research activities. In addition to courses on numerical mathematics he presents lectures on mathematical methods in fluid mechanics and mathematical modelling and supervises seminars on continuum mechanics and numerical mathematics. He has significantly contributed to the development of numerical analysis and mathematical modelling at the Faculty of Mathematics and Physics. His lectures are highly evaluated by students. He has trained a number of graduate and doctoral students who were awarded in various national and international competitions. Many of his former students became prominent experts, associate professors and full professors at universities in the Czech Republic and abroad where they continue his work and fulfil his aims and objectives. In the years 1994–2012 Prof. Feistauer was member of the Scientific Council of the Faculty of Mathematics and Physics. He has worked in various committees and bodies of the faculty, but also beyond. He is member of the committee for doctoral studies F11 and M6 at Charles University, he was member of the Scientific Council of the Faculty of Mechanical Engineering, currently he is member of the Scientific Council of the Faculty of Chemical Engineering at the Institute of Chemistry and Technology in Prague and for many years he worked in an advisory commission of the Grant Agency of the Czech Republic. He is member of numerous scientific societies (GAMM, ISIM, AMS, EUROMECH, ECMI and others) and member of the editorial boards of five international journals. His research and teaching activities were awarded by the medal of the Faculty of Mathematics and Physics and by the silver medal of Charles University. In 2004 he was elected member of the Learned Society of the Czech Republic. In 2006, Prof. Feistauer received another prestigious award, the title of Honorary Doctor of the Technical University of Dresden. Among his more recent awards, Feistauer received the 2008 Memorial Medal of the Charles University and the 2013 Memorial Medal of the Czech Mathematical Society. His nonprofessional interests include the history of art, music: painting, playing violin, viola, and music composition.