There are various ways to enter accented characters, one of them is using html.
For example, I can type Gâteaux by entering
Gâteaux; I can type entering François by entering
François. Will search find posts using either of them if I search for Gateaux or Francois?
I should mention that this question was inspired by a recent discussion on MathOverflow: Formatting: Diacritics outside of Formulas (The same meta also has a closely related older discussion: LaTeX style accents in regular text.)
I have posted an answer in the sandbox to be able to test this here. (The answer contains only "html-versions" of Gâteaux, François, Gödel, Pólya, Čech.) They all seem to be working correctly with the exception of Čech.