The wikipedia article on Method of mean weighted residuals has a section on choice of test functions, in particular it says about Galerkin method:
The Galerkin method, which uses the basis functions themselves as test functions or in the more general case of a nonlinear assumed form (where the nonlinearity is in the degrees of freedom) of the solution the Galerkin method uses the test functions: $$ w_{i}={\frac {\partial u}{\partial a_{i}}}$$
I'm really interested in the "nonlinear assumed form". I would like to know what is known about Galerking when we assume nonlinear form of the trial function.
Unfortunately the wiki article gives only one reference to a book which I was unable to get hold of. Can anyone direct me to any other source where I can read more about this topic?