Abstract
The origins of the alternating Schwarz method lie in the difficulty to prove the Dirichlet principle. This principle was evoked by Riemann in the proof of what is now the well known Riemann Mapping Theorem. We tell in this short paper the story of this exciting journey through the world of research mathematicians, up to the first computational Schwarz methods.
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Notes
- 1.
The centripetal force is inverse to L × SP 2, it is inversely proportional to the squared distance SP. Q.E.I.
- 2.
“A teacher, Professor Schmalfuss, lend him Legendre’s book on number theory, a very difficult work of 859 pages in quarto format, and he got it back already after a week. When he tested Riemann in his final high-school exam on this subject much more thoroughly than usual, he realized that Riemann had completely mastered the content of the book.”
- 3.
The manuscript submitted by Riemann is a testament of the thorough and deep studies by the author in the area to which the treated subject belongs; of an aspiring and truly mathematical research spirit, and of a glorious, productive self-activity. The presentation is comprehensive and concise, partly even elegant: the major part of the readers would however in some parts still wish for more transparency and better arrangement. As a whole, it is a dignified valuable work, which does not only satisfy the requirement one usually imposes on a manuscript to obtain a Ph.D. degree, but goes very far beyond.
The mathematics exam I will do myself. I prefer Sunday or Friday, and in the afternoon at 5 or 5:30 pm. I would also be available in the morning at 11am. I assume that the exam will not be before next week.
- 4.
Two simply connected surfaces can always be mapped one to the other, such that each point on the former moves continuously with the point on the latter….
- 5.
Weierstrass had taken Riemann’s PhD thesis as vacation reading, and complained that for a function theorist like him, the methods of Riemann were hard to understand. Helmholtz then also borrowed the thesis, and said on their next meeting, that for him, Riemann’s thoughts seemed to be completely natural and self-evident.
- 6.
To this end, one can often invoke a principle for finding a function that solves Laplace’s equation, which Dirichlet has been using in his lectures over the past few years.
- 7.
Dirichlet’s reasoning apparently leads to an incorrect result in this case [8].
- 8.
…my existence theorems nevertheless hold [8].
- 9.
For us physicists the Dirichlet principle remains a proof [8].
- 10.
The method of conclusion, which became known under the name Dirichlet Principle, and which in a certain sense has to be considered to be the foundation of the theory of analytic functions developed by Riemann, is subject to, like it is generally admitted now, very well justified objections, whose complete removal has eluded all efforts of mathematicians to the best of my knowledge.
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Gander, M.J., Wanner, G. (2014). The Origins of the Alternating Schwarz Method. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_46
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