From Euler, Ritz, and Galerkin to Modern Computing

The so-called Ritz-Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in Hilbert's "direct" approach to the variational calculus of Euler-Lagrange and in the thesis of Walther Ritz, who died 100 years ago at the age of 31 after a long battle with tuberculosis. The thesis was submitted in 1902 in Gottingen, during a period of dramatic developments in physics. Ritz tried to explain the phenomenon of Balmer series in spectroscopy using eigenvalue problems of partial differential equations on rectangular domains. While this physical model quickly turned out to be completely obsolete, his mathematics later enabled him to solve difficult problems in applied sciences. He thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics. We will see in this article that the path leading to modern computational methods and theory involved a long struggle over three centuries requiring the efforts of many great mathematicians.